Firstly, if the examples that come with the engine are working correctly then you have NOT installed Tomcat with fault. Good. To run one of your servlets you must compile the code and place both the class and the source file under webapps\Web-inf\classes
Once you have done this edit the web.xml file sitting in the web-inf directory. The xml file should register the servlet name, the name of the class (of the servlet) file and the mapping.
Use the example web.xml file supplied to understand how it works.
You said it was running as a standalone. So what has apache got to do with it?
All the best