Just guessing here. The school may be using the same tech that corporations use. They have locked a profile to the computer from the factory using DEP. That profile has restrictions that do not allow app installation. There is nothing you can do to bypass this type of protection. Even wiping the computer will not help as part of the restart will reinstall the MDM restrictions profile.
You should not be able to unlock websites either. You can always use a proxy to bypass school filters, but this would be in violation of federal law as well as school policy. This is pretty obvious on the network, if anyone is watching. I wouldn't play the cat and proxy game, I'd build a whitelist.