Mixing the desktops and cluster doesn't sound like a good idea. I'd expect it to create a lot of admin issues especially in a high use teaching lab.
I'd recommend minimal systems for desktops, PXE boot from the network, and create a cluster separately. This should be a low admin solution as a reboot of the desktop will load whatever the new OS changes are. If you're using, for example, gridengine, make the desktops clients and people can qlogin or qsh to the cluster for interactive sessions, also have their home dirs and data dirs automounted locally when they login. As you say, if there are NFS issues everything goes down, but that's an issue anyway in a networked environment. A mid-way alternative is something like the Rocks distro which (if I remember correctly, haven't played with it in a while) will update a local OS install on boot.
In terms of installs, if it's a teaching lab you really don't want local installs of software. What we do on our cluster is have a /data/binaries directory where the shared binaries are installed, that works fine.
If you want to go very cheap on the cluster hardware, a friend who has assembled many machines has a successful and simple heuristic for choosing good cheap hardware; go to to Tom's Hardware (or other site), for each component you need, choose the cheapest of the top 5 or 10. This gives a good price / performance ratio. Given the budget your cluster will not benefit much from rack mounting as there will be only a few compute nodes, and you pay a premium for rack hardware vs beige boxes.
Note that cheap hardware can be noisy, particularly under high load as the fans really spin up; this may be a consideration if the cluster nodes are being housed in the same room as the desktops.
You'll probably want a local switch for the cluster and desktops given the number of nodes, 32 ports at least (desktops and cluster interconnects). You may want to talk to your institute IT about the interconnects between your network and theirs.
NAS are cheap for storage, and a system like OpenFiler with commodity hardware can get you 24-48T of RAIDable storage at a reasonable cost. If you just buy a commodity box and configure it yourself, setup and maintenance may be expensive time-wise depending on your level of knowledge. I've had good experiences with the more expensive pre-configured appliances (ReadyNAS, and others) and though the performance isn't great, they're very reliable and almost zero maintenance.
I would strongly recommend having some sort of backup mechanism; RAID is not backup. Either tape or another NAS holding backups for key stuff.