Thank you for your post.
Unfortunately, it is difficult to determine what is causing the problem, but here are some steps that may help you in the short run.
Launch Terminal (under Applications->Utilities) and enter the following commands:
fmsadmin close (this will close all of your database files)
fmsadmin stop server
fmsadmin start server
When you lose connection to the server, is it just your web connections? That is, can other FileMaker users still open the database files? I'm trying to determine if just the web publishing engine has stopped or all of FileMaker Server.
Tomcat is used to communicate between the server and web publishing engine, so the key may be in those log files. They are located at:
FileMaker Server -> Admin -> admin-master-tomcat -> logs
Any other information you can provide may be helpful in narrowing down the possible causes.