Generally speaking, web applications are referenced by the first item after the domain. I think you're in for a world of hurt if you try to 'shoehorn' applications into the same directory.
http://ourfineschool.edu:8088/examples I would take 'name_of_app' out of the examples app, and refer to it like:
http://ourfineschool.edu:8088/name_of_app http://ourfineschool.edu:8088/name_of_other_app This is a highly recommended way of doing things.
servlet.jar -> contains the servlet API classes (like HttpServlet and HttpServletRequest). There should only ever be a single instance of this jar file (in common/lib). As for 'different' servlet.jar files... your developer should not be giving you one. It's not required. There are different versions of the servlet spec, for which there exist different servlet.jar files.. but as long as your app was compiled with version 'n' of servlet.jar, then a jar file of n+x should work (but not n-x). This is standard "backwards compatible" behaviour.
As for
"When I try to add information for the new app it "breaks" the server for the old app." The 'server' is configured in CATALINA_HOME/conf/server.xml. Look in here for <Context> tags. Each <Context> will equal one configured application. And yes, the /examples one should be there. ROOT *is* the default app, which is why you can just say "http://ourfineschool.edu:8088/" and get that app (you get the default when you don't specify the web-app in the browser).
Tomcat is smart enough to 'autoconfigure' any application that is dropped (in WAR format) into its CATALINA_HOME/webapps directory. It will also autoconfigure any unpacked directory under webapps. One requirement here is that the directory (and the WAR) must contain a non-empty WEB-INF/web.xml file. A web.xml file with nothing but <web-app /> should be enough. For auto-configured apps, the name of the app is the name of the folder or WAR file (without extension).
Hope that helped, but ask more questions if not!