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serving static content and java web apps from one server

I'm trying to build my own site where initially I will only host my profile and resume but will also build and publish some simple web applications that I can use as coding samples for jobs. The idea is people can see a demo of the apps and then see the code on github.
Can I do all this using only tomcat or should I use another server for this purpose?
I have heard that tomcat should only be used for java web applications but I've tried it locally and it seems to work...

You can totally do it all with tomcat.
Arguably, tomcat is not as efficient as apache or ngnix for serving static assets, 10% less so, and does not scale as well. There's a good article on the subject here:
http://tomcat.apache.org/tomcat-3.2-doc/tomcat-apache-howto.html
This is not not going to be an issue for you with a small site. Go with what gets you up and running the fastest.
If you want, later you can show off your technical prowess by running tomcat behind a web server. You can tell folks at interviews the story.

Which one is the better configuration: Tomcat only OR Apache Server + Tomcat?

I have a VPS where I want to install a web application for production.
Currently, it works on Tomcat 6 connected through Apache Server using AJP module.
Every thing works ok, but I want to know if it is a good idea to configure Tomcat as the main web server due we gonna run only JSP applications.
FYI: the server must handle https requests.
Thanks in advance.

Misinformation
Years ago there was much misinformation against using Tomcat directly as a web server, but in fact Tomcat works very well as such.
I've used nearly every version of Tomcat on various projects, though all were relatively low volume. Never let me down – well, the WebDAV module never worked right. But other than that Tomcat is fast and reliable both for static content as well as dynamic servlet work.
httpd
Apache HTTP Server ("httpd") is of course very advanced and full of features. If you need those features, then use httpd. If you have very high volume traffic, then use httpd. If you have huge amounts of static access and want to relieve Tomcat of that duty thereby letting Tomcat focus its performance on your servlets, then use httpd. If you want to keep a front-facing web site running while take Tomcat up and down, use httpd. But for most relatively simple web sites or simple web apps, Tomcat suffices.
I suggest you try Tomcat alone and see how it goes. Start with your development and testing systems. Review your config files for httpd to see if all its features can be reproduced using Tomcat features. If that all works, then do some load-testing. Either hack some load-tests of your own, or try any of the many load-test frameworks.
While you can hot-deploy webapps to Tomcat, that was not always recommended in production (I'm not sure about current versions). Plan how to handle taking down Tomcat to deploy updated apps and to perform maintenance chores.
Recent Tomcat
You say you are using Tomcat 6. You may want to consider moving to Tomcat 7 or 8, but review the release notes.
Jetty
Also consider Jetty, the most direct competitor/alternative to Tomcat. Jetty is much akin to Tomcat in its purpose, scope of features, and great reputation.

If you're using SSL you will get the following benefits by front-ending with Apache HTTPD:
Ability to specify cipher suites in order of preference, which is practically mandatory if you want case-hardened SSL, and impossible with Tomcat alone.
Ability to request or require client certificates on a per-location basis instead of globally, which is Tomcat's only mechanism.
There are many other benefits as well, such as:
the ability to load-balance between multiple Tomcat instances, which alone is probably enough of a reason to do it
more control over what's logged and where the log files go
more tools to defend against attacks of various kinds.
I used standalone Tomcats for years, but having made the switch I would never go back even for a clean-sheet project.

I don't think either is better, it depends on your requirements. In environments where you may have to address security or logging requirements horizontally across heterogeneous backends (IIS, Tomcat, etc), Apache is your friend (or an expensive load balancer).
Assuming you don't have these requirements, I don't know of any advantage in using Apache for Apache sake.
SSL is easily configured in Tomcat and performance these days is likely to be on par with Apache and can be improved with APR.

Java design question on which framework to use? JBoss, Rest, etc

I'm interested in creating a client/server app in java, but not certain which framework to use. Here are simple scenarios to check disk space:
Scenario 1
1. A user would click on a button on a web page console "Get current disk space usage".
2. The servlet would get all machines from a db and ask each one to return their used disk space.
3. Each machine would reply with the used disk space.
Scenario 2
4. Each machine would update a service every 24 hours with its used disk space.
From what I can tell, I need to have a tomcat instance to host a servlet as the console.
From here is where it gets murky for me - and hence this question.
Would I also need
a) a SOAP service for #4?
b) a SOAP "client" running on each machine for #2?
I'm just not sure when to use SOAP services vs. RESTful services, etc.
Any opinions would be appreciated.
Fairly new to java, so apologies for any mis-spoken jargon.

You don't need a SOAP service nor a client. It would be nice, of course, but (if in a hurry) you can use your own format. CSV's, pipes, whatever fits you.
I've found SOAP services are easier to read but they are not the best choice when you expect a lot of data, mostly 'cos of the tags that you have to open and close. JSON is my choice in these situations.
Now, what you're talking about are not frameworks. They are protocols or convention. A framework is more language-dependent, like Struts, Spring, Tapestry, iBatis, etc.
Hope this helps.

SOAP is only relevant in context of an open ended system where service discovery, etc. would be useful. Otherwise, it is a PITA and entirely overkill.
RESTful services -- recommendation: use json -- would more than suffice for your purpose.
Tomcat is not your only option. Jetty is quite excellent as well.

I would suggest lightweight approach, as your use case is fairly limited, I would suggest going with embedded Jetty, Spring MVC and possibly JSON or XML response.
This will simplify deployment as you can wrap Jetty in wrapper application and make it runnable as process or service in either Windows or Linux, plus the footprint on the whole system will be fairly small.
This could change if you plan to extend this application with more feature and more integration point with other services/applications/protocols.

Benefits of Tomcat (or equivalent) for a simple service

I'll need to develop a Java service that is simple because:
It only communicates via a TCP socket, no HTTP.
It runs on a dedicated server (there are no other services except the basic SSH and such)
Should I make this a standalone service (maybe in something like Java Service Wrapper) or make it run in a container like Tomcat? What are the benefits and detriments of both?

If you aren't working with HTTP, you will have to build your own connectors for Tomcat. When I've written these types of applications, I've just written them as standard Java applications. On Windows machines, I use a service wrapper that allows them to be part of the Windows startup process. On non-windows machines, you just need to add a start up script.

Using a container (regardless which) buys you that all the details about starting, stopping, scaling, logging etc, which you have to do yourself otherwise, and it is always harder than you think (at least when you reach production).
Especially the scalability is something you need to consider already now. Later it will be much harder to change your mind.
So, if somebody already wrote most of what you need, then use that.

Tomcat doesn't sound like a good choice for me in your situation. AFAIK it's primarily made for Servlets and JSPs, and you have neither. You also don't need to deploy multiple applications on your app. server etc. (so no benefit from ".war").
If you need dependency injection, connection pooling, logging, network programming framework etc., there are a lot of good solutions out there and they don't need tomcat.
For example, in my case I went for a standalone app. that used Spring, Hibernate, Netty, Apache Commons DBCP, Log4j etc. These can be easily setup, and this way you have a lot more freedom.
Should you need a HTTP server, maybe embedding Jetty is another option. With this option too, you have more control over the app. and this can potentially simplify your implementation compared to using a tomcat container.

Tomcat doesn't really buy you much if you don't use HTTP.
However, I was forced to move a non-HTTP server to Tomcat for following reasons,
We need some simple web pages to display the status/stats of the server so I need a web server. Java 6 comes with a simple HTTP server but Tomcat is more robust.
Our operation tools are geared to run Tomcat only and standalone app just falls off radar in their monitoring system.
We use DBCP for database pooling and everyone seems more comfortable to use it under Tomcat.
The memory foot-print of Tomcat (a few MBs) is not an issue for us so we haven't seen any performance change since moved to Tomcat.

A container can save you from reinventing the wheel in terms of startup, monitoring, logging, configuration, deployment, etc. Also it makes your service more understandable to non-developers.
I wouldn't necessarily go for tomcat, check out glassfish and germonimo as they are more modular, and you can have just the bits the need, and exclude the http server.
We faced a similar decision a while back, and some parts of the system ended up being jsw based, and the others as .war files. The .war option is simpler (well more standard for sure) to build and configure.

Which Java EE server should I use?

OK, this is a little unusual, because ANY Java EE container can do the simple things I want to do (XML processing, JPA, Hibernate, SOAP/REST web services, etc). This is for personal use, more to gain skills than to accomplish essential functionality. I have my own Linux server (Ubuntu Jaunty x86_64) with business class internet, so I can install pretty much anything.
I use Tomcat a lot now, however, I ran into a few situations lately job hunting where they were looking for experience in a specific Java EE container (which defeats the whole purpose of having a standard), and of course not the same one each time.
So what I'm looking for is a Java EE server that:
Is in demand in the marketplace right now
Is free (or
Is not too horrendous (disk space, time) to install and to deploy applications on
Runs on Ubuntu x86_64
I've been able to glean some information via Indeed's keyword search, and that tells me to stay away from Jetty/Glassfish, even though they fit "lightweight" and "free". I also see from this SO post that WebSphere is a steaming pantload of bloatware that's hard to deploy/configure, but I don't know if that's accurate or current. I like Tomcat (completely FOSS, smallish, easy to deploy, plenty of docs/users), but it's less in demand than some of the bigger boys.
So what would you recommend I install? Thanks in advance.

My personal recommendation is JBoss. Not only is this a solid server, but there are other JBoss Platforms and Frameworks that easily integrate into JBoss. One example is JBoss ESB.

Far too often, I've seen servers like JBoss and WebSphere (I've worked with them for the last 5 years now) rush to market with broken implementations. Unfortunantely, the part that is broken might be very small, but it will be just what you will be breaking your head on.
Oracle Weblogic just kicked the hell out of me with a broken implementation of MDBs (annotations).
If you want a server that is the most Java EE 5 compliant, will give you the least heartache and pretty much production-worthy, go for Glassfish and yes... use Netbeans as your dev IDE.

+1 on JBoss, that is mainly what I use for personal use and work. I have also used OC4J, and find that it is harder to use, and more finicky with it's class loading mechanism, maybe I'm just used to JBoss. But there is definitely more and easier to find community support for JBoss. So that is reason enough if you are doing this for personal use and are not going to get a support license, there will be many more users willing to help out if/when you encounter issues.

I've only used Tomcat myself, professionally. I've been using it for the better part of my (so far) three year career and I havn't experienced many problems with configuring or deploying applications to it. I use Eclipse WTP for Java EE developers 3.4 (Europa) at work, and mostly develop JSF applications utilizing ICEFaces. We deploy using the Tomcat web manager for most of our smaller projects, and through Hudson for some of our broader applications.
Please bear in mind some caveats, however:
We host about 20 seperate web applications, mostly JSF with a few older JSP apps.
The web applications served by my office have a very, very narrow scale. Our department currently supports about 20 unique users, so most of our applications do not receive a lot of heavy traffic.
Most of our web applications are very processor intensive (i.e. here's a 2GB CSV file...write a program to calculate the total number of people that reside within homes that have 3 or more children...) that only usually perform one-off runs either weekly or daily (or even yearly...). In other words, "Give me all your resources for 2 hours...then I'm going to lunch".
Our primary web server is a Solaris 10 box with a Dual core sparc processor (I believe clocked at 1.8 GHz if memory serves correctly), with 2GB of RAM (don't recall if it is DDR or DDR2).
That being said, I've definitely noticed performance problems over time with regards to our application environment that probably have more to do with our application code as opposed to Tomcat itself. I'm willing to bet that the problems have to do with poor memory management within our processing logic and data allocation than anything. The reason I believe this is because on a fresh restart of Tomcat our applications usually execute in an incredibly responsive fashion, yet over the course of even just one or two days the user experience is degraded.
Take it for what you will. We've probably got some problems to deal with on our end, but Tomcat itself seems to be pretty performant in a localized environment.

I ran into a few situations lately job
hunting where they were looking for
experience in a specific J2EE
container (which defeats the whole
purpose of having a standard), and of
course not the same one each time.
Well, I think you have (half) answered the question yourself! If your primary aim is to help you find a job, you need to look at what Java EE container the recruiters in your geographical area are asking for. Go to your favourite recruiting websites, pull up the Java / Java EE jobs for your area for the last couple of months, and count for each kind of web container. (And also count the jobs that don't specify a particular web container.)
If the results of your research say that you need to learn WebSphere, you have my sympathy :-)
Don't forget that a "mandatory" requirement may not actually be mandatory if you are you are good enough in other areas. It all depends on what other candidates the recruiter can find. Don't be afraid to say "you asked for X, but I have Y and Z which demonstrates that I have broader experience with this technology".

Oracle officially ceased to support Glassfish. All bugs found there (and there are quite a lot of them) will never be fixed.
Payara is now the best option, as this is a clone of Glassfish, but is continued to be developed by the Java community.