Yes, you do need a plugin. I recommend the sysdeo plugin, as it allows full access to Tomcat's features.
However, there are Eclipse distros that are pre-bundled with other ways of running Tomcat, as well. The
J2EE spins of Eclipse come with the WTP plugin, which can be used to launch, control, and debug in Tomcat, although it mangles the runtime enviroment in ways I find very annoying.
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.