What is / how can I get a student-safe power supply for electronics projects?
I know you said you don't want to run outlets everywhere, but I think a pile of wall warts is exactly what you want.  Jameco has a large selection.  9 V sounds pretty good and that is one of the standard voltages, but you might want to consider 5 V (see below).  Get something in the 500 mA to 1 A range.

These supplies are inherently isolated from the line, are usually short-circuit protected (check to make sure, you definitely want that), and draw so little AC power that you can string a bunch of outlet strips together without harm.

In the end, I think this will cost less and provide a better experience for the kids.  I remember when I was a kid tinkering with this stuff how frustrating it was to have batteries run down, especially when you're not aware of when you are asking a lot from them.  You also feel a lot less guilty abusing a power supply than running down consumable batteries.

You can start a fire with almost anything.  If you do just the right thing, even a 9 V 500 mA supply can catch something on fire, but no more so than a 9 V battery and without the chance of the battery itself doing something bad and causing chemical burns.

If you are worried about LEDs getting damaged by them getting hooked up backwards, maybe you should get 5 V or 6 V supplies.  Most LEDs can handle 5 V in reverse, and some 6 V.  With 5 V supplies, you can eventually run logic cicuits directly without having to use a linear regulator.  Lots of stuff will run well from 5 V.  If you get 5 V supplies, get at least 1 A capability.  That will be useful for running small motors.  5 W total power really isn't all that much.