Fun questions! I love Eureka, so I'll see what I come up with for that third questions.
Power Level: I would aim for 20-25th level for the "full professors" and 10-15 for the "staff and assistant professors." Some departments could be lacking full professors for some reasons, and left with just staff. What happened to the full professors could be adventure fodder. Part of my reasoning on these levels is, if they're teaching or doing research, they need enough PP to cast the spells they would be using every day, or on "big project" days. (I'm sort of assuming many/most faculty will be casters, but that's really a bad assumption
)
Departments: I think your starting list is good. Rather than giving you a list, I offer two guiding ideas.
(1) What classes of faculty do you want to have? Those can then specialize in spell lists or skills that help define departments. Start with this and then add some more esoteric departments (Perfume making...Essence Dance...High altitude athletics...)
(2) Your numbers give an Academy staff level of 400-600, and that staff has to have a town that supports it, or in-house resources. So they need food, sanitation, paper/research supplies, beasts of burden, etc. You can build much of this into the departments and their staff, as well as their daily routine. Animists who tend to the livestock used to work the Academy fields for food. Alchemists, both magical and mundane, who make the paper, ink, reagents, crucibles, whatever. Engineers from the engineering program who use "Wall" spells and mundane methods to build and repair the campus. Not every department has to contribute to the Academy's operations, but it may help add some less obvious departments and flesh out how your town runs.
Weirdness: There are always explosions of unknown origin. Magical constructs and golems going out of control. Weather spells gone awry. Combinations of those, so maybe a weather spell crosses with a golem construction and the golem is inert but the creator has control of a cloud but doesn't realize it. Or the golem throws lightning but the creator can't control that part. Throw in an NPC who's brilliant but socially clueless and tends to fumble the fix a few times before getting it right. Invisibility pockets that drift through town and wreak havoc. Strange food resulting from the agricultural experiments. Smoke coming from an unknown source - where there's smoke but no fire. Then later they find out the fire was in a totally different place. Someone's "human kindness" weak charm spell being more powerful than intended and resulting in sudden romances or outright debauchery in the streets. (Or same spell leading to excessing politeness like the chipmunks in the Looney Tunes' Goofy Gophers. Nothing gets done because everyone is stuck in a "no, by all means, after you" loop.)
