attaching the RAM suction mount to the glass windshield or window
I suggest that you have the permission of the person who owns the plane (not just the instructor, but the owner). I do not allow headsets on the glareshield, for not wanting scratches on the inside of my windshield, so I certainly would not be sticking things to the inside of the windows. I agree that the suction cup itself will not scratch, but dirt, or if someone wipes off the smudge after, may result in a scratch, and you don't want to know the cost to replace a windshield. As said, if to stick something on a window, in your plane of view, you're obstructing outside vision, I would not allow that.
If you're attaching something behind you, make sure that it cannot come off and hit you during the crash. To meet safety requirements, it must be able to withstand a 9G times 1.5 factor of safety force. It is very unlikely that your instructor is qualified to make this assessment, seek the concurrence of the aircraft maintenance person.
PA-38's are lots of fun! They're crisp to fly, and worth the effort in precise learning. Some pilots nay say them, but I flew a couple of them for a few hundred hours, and quite liked them.