there are no problems using different versions of animate (though if you use a feature available in one version that's not available in another, that feature won't be present and you'll see that warning).
and you can install older versions along-side the current version if that presents a problem.
To add a few specific details:
Flash Pro CS5.5 and later can open any FLA that was made in CS5.5 or later. If school has CS5, you can open the files at home, but wouldn't be able to reopen the files at school. CS5 is quite old, so hopefully they will be on something more recent.
Some features came and went, and came back again! TLF text fields is one notable example, if school has CS6 for example it would be very easy to accidentally make TLF instead of a regular textfield. Watch out for that.
Inverse Kinetics went away for a while, then came back. I haven't tested to see if CS6 IK is ok in the latest Animate.
CS6 and earlier could make ActionScript 2 FLAs, and those don't work in later versions. Make sure you're set to ActionScript 3.
I'll be using ActionScript 3 both at college and at home, thats without a doubt, but with the rest of the stuff, I've got no idea.
it’s unlikely you’ll have a problem.