" . . . what device is best to use . . .?"
Does not matter as any mac will do. IMO the answer to your second question rules out a laptop as it would be a poor use of an expensive portable device more suited for a different purpose. Most people in your situation tend to go for a MacMini. It's cheaper than the other two and has a very small form factor. Ideally buy two and set them up in a redundant configuration.
"Does the device need to be on all the time with the same network address?"
As you know Apple's MDM is called Profile Manager. You get Profile Manager with OS X Server. OS X Server is an App and not an OS. That said it should still be treated as a Server with all that that means. A Server should have a fixed location within a network it resides in. It should be accessible by the devices you're trying to manage which implies a fixed IP address and always on availability. Beyond the fixed IP address nothing will work well (if at all) without a properly defined hostname and a DNS server that can resolve that name properly. This can't be emphasised enough and is crucial. Get that part right at the beginning and everything else will 'just work.' Your school's IT Management should have already done this anyway although depending on their understanding of what DNS is it may prove a potential stumbling block?
You can access the PM web-based management portal from anywhere you like. However this will mean configuring whatever firewall the school is using. For example allowing VPN access from whatever computer you're using at your home - Mac or PC. Does not matter which as it is via a web browser and any web browser will do - or configuring the school's external domain to do the same.
You should also look beyond PM. PM won't back-up the devices for you. If iOS devices are involved you should consider using a combination of iTunes and/or iCloud and/or Apple Configurator to back-up any locally stored data. I would steer clear of this and use something like WebDAV Navigator or Foldr instead.
For OS X devices network logins would be the way to go. Student data would be stored on a drive attached to your Server. This data would in turn be backed up by whatever backup solution the school is using. Does not matter whether the network is based Microsoft's Active Directory or Apple's Open Directory. I would also consider using DeployStudio, Munki and Apple Remote Desktop for a more rounded management/deployment approach that go far beyond what is available in PM alone.
It can be simple and straightforward and in practice it is as long as you know what you're doing and where the potential gotchas are. You may benefit from an experienced consultant coming in and advising/assisting you in the initial setup? It makes sense as using Apple devices in an enterprise network can be painful if the initial 'get-off-the-ground' phase is handled poorly.
My 2p.