Tomcat "Just Works". You should not normally have to do anything to it on a routine basis as long as the webapps are well-behaved. And if they aren't, whatever maintenance is required will be determined by what sorts of misbehavior the webapps are indulging in.
The only thing that isn't self-cleaning that isn't up to the applications would be the Tomcat logfiles, but if/when you have to stop Tomcat to clean them out depends on what the applications are sending to console logging. Tomcat does log a few things itself, but if no apps are contributing, it would be a long time before you'd see enough written to the log to warrant stopping Tomcat to clean the logs.
Customer surveys are for companies who didn't pay proper attention to begin with.
Personally, I use the Management app once a day to check the various statistics such as number of requests, number of errors, bytes sent and especially the condition of the request handling Threads. If an application is hanging somewhere you will see request threads with very long running times.