That sounds very frustrating.
When I think of things that can go wrong on the server and that a reset fixes, I think of two things:
Server scheduled scripts that hang or run away, and/or Perform Script in Server hanging up. Both use the SASE part of FileMaker Server which web publishing does not: which could jive with your experience of Web Publishing staying up.
You can see the state of your scheduled scripts in the admin console. If they all look good (have realistic last-run dates and times) you can turn perform script on server (PSOS) OFF in the calendar. If you're using DayBack Calendar (our latest), instructions for turning PSOS off can be found here:
http://www.seedcode.com/pmwiki/index.ph ... Maker.PSOSAs to *why* your scheduled scripts might hang up or why SASE could stop responding, FileMaker Server doesn't give us a lot of visibility into that. FMS 15 is a big improvement in some areas so if you're looking for a reason to upgrade, perhaps this is it. You could also look at your hardware, etc. but I'm not an expert in tuning those things and tend to rely on my hosting providers.
Let us know what you find.