Imessage on school devices

Just about any educational institution will have some policy/policies applied to their devices. One of the most common being restricting signing in with personal AppleIDs. That said no-one here can really help you as this is a question for your school IT department. You must remember this is not your device, does not belong to you and should only be used for its intended purpose. If you want to use iMessage with your AppleID use it on your own personal device assuming you’re allowed to use it during school hours.

Remotely, likely not. If you hand them your device, likely so, as would be possible if you hand anyone your device. The best advice is to treat a device that is owned and issued by your work/school as their device and not yours and to not put any information you don't want them seeing on the device.