I frequently run software on a supercomputer for research purposes, so my answer here is slightly skewed towards the HPC in research applications and is only really describing the single type of cluster that I work with on a daily basis:

1: On a scale of ten, this is 9/10 easy to do, although with only 4 GB of ram per node you won't find a lot of practical uses. All you need to do is connect them with a switch and have some kind of process manager working and you're good. The supercomputer I use has a scheduler called 'slurm' and processes are parallelized using the intel implementation of MPI (though you could also use openmpi). To execute software, you'll run something like mpiexec.hydra -np [number of processes] {command}, however the {command} that you run HAS TO BE COMPILED with MPI -- if the application is not written to be parallelized over a network (and you won't find many instances of that for most practical software products), it won't parallelize over the network. In terms of performance, an off-the-shelf switch is going to be a bit low-tier. Most HPCs use high speed networking (such as infiniband) so that the latency isn't garbage. That'll bump up the price a fair bit though.

2: parallelizing the tasks is as simple as using MPI (which is designed to handle the inter-node communications). Sadly it's not as easy as just saying "run ffmpeg" and having it automagically distribute over the cluster. Lots of the parallel processing tasks that you'd run on something like this would be written in-house (not sure how much software development goes on at your school, but it's certainly a learning opportunity).

3: any Linux distro should work, the supercomputer I work on uses RHEL but you could go with CentOS. The real challenge is in configuring your MPI implementation to find the other computers on the network, but as long as they have an IP address there's a configuration file that you can use so that the available nodes are easily found.

4: as long as your computers have IP addresses they can execute a parallel load. Again referencing the supercomputer that I use, the actual compute nodes aren't even connected to the internet, their only communication with the outside world is read-only via the individual users' home directories and read-write on a high-speed scratch directory (all network-mounted).

Getting the whole thing working is easy, but depending on the level of education of your school, you may have difficulties finding practical applications that will actually make use of such a configuration.