For instance, the new multimeters need a 500 mA fuse. But is that a swift, slow-acting or half-slow-acting fuse (couldn't find the proper translation for that one)? Does it matter?
Yes, it does matter. Measuring current with a multimeter is actually a difficult job for a multimeter. That's why most multimeters either don't have a current range or it's very limited - miliamps on a multimeter that can measure hundreds of volts.
You need to teach your students what a difficult job this is for the equipment. The way to to this is to get the fastest, most sensitive fuse you can. This should protect your expensive equipment from students and when it does blow (mutiple times in one class) then the students will learn to be very careful with a multimeter set to measure current.
Do NOT keep the spare fuses in the classroom. Make sure the student has to walk to the supply store which is at least down the end of the hall, if not in the next building. By making the student who blew the fuse do this walk of shame, you will burn it into their memory and they may eventually become a good electrical engineer who understands how fragile the current setting is on a multimeter.
Of course you get to use the good equipment with current shunts and sensors that don't blow out with too much current when you do your demonstrations at the front desk. You don't want an accident causing you to send a student out for a spare fuse and make the rest of the class wait.
Transformer demonstrations? I never got that in school. The teacher just gave us the transformer ratio equation and just told us that's how it works - no demonstration to prove the teacher wrong with those pesky geometry constraints.