You'll need a 10/100 switch, or ideally, a 10/100/1000 switch. If you have 15 computers, I would recommend a 24 port switch. You never know what other peripherals you might want to add (such as a printer, or more computers.)
The Time Capsule does not need to be online to do backups, it just needs to be on the same network as your computers.
OS X Server software costs $250 for schools. It will run on any Intel Mac.
OS X Server seems like overkill. Can anyone else suggest an alternative?