Mac server keeps "disappearing" and "reappearing" on local network
The Finder uses mdns (bonjour) to populate the list of servers in the sidebar. You can look at the advertisements directly using:

dns-sd -B _afpovertcp


This will show you all the afp fileservers on the local subnet. You can replace the _afpovertcp with _smb to browse for windows fileshares.  Run this command for a while and look to see if there are any "Remove" messages mentioning the file server. 
Time machine is likely using something similar to:

dns-sd -L servername _afpovertcp


to contact your server. 

If you see a bunch of similar names with a 1 or 2... at the end, you may be seeing the server reregistering its name for some reason and not being able to get its old name. This could be due to some other network machine running a sleep proxy, and not releasing the name when it should.