It is definitely possible to control start and stop of your Tomcat server. If you think about it, starting and stopping Tomcat is mostly done via the script files (start.bat, stop.bat) provided by the Tomcat binaries. You could tie this script to any process, one of which could be your web application!
SCJP 1.4, SCWCD 1.4 - Hints for you, Certified Scrum Master
Did a rm -R / to find out that I lost my entire Linux installation!
That would depend on what your hosting service supplies. Some ISPs provide cPanel. WebMin is also popular. But there are others as well.
Generally, to control Tomcat, you'd have to have a Tomcat plug-in installed.
The secret of how to be miserable is to constantly expect things are going to happen the way that they are "supposed" to happen.
You can have faith, which carries the understanding that you may be disappointed. Then there's being a willfully-blind idiot, which virtually guarantees it.