9 speed derailleur on 7 speed cassette? I currently have a 7 speed cassette, 7 speed derailleur and 7 speed shifter from Shimano. Unfortunately, my rear derailleur (Shimano Tourney) broke, so I want to replace it with something better. The problem I'm facing though is that I'm finding only 9 speed derailleurs that fit what I'm looking for and where I don't have to pay stupidly high shipping fees (I was looking at the Shimano Alivio line).

That's why I'm wondering if a 9 speed derailleur works with a 7 speed cassette and shifter. Would it work with the chain I have right now or does a derailleur only work with chains rated for the same amount of gears?
Yes, all 9-speed Shimano rear derailers share the same actuation ratio as all Shimano 7-speed ones,  except for pre-7700 Dura Ace.

I've never seen a 7-speed chain rub inside a 9-speed RD cage. If somehow that were an issue, an 8-speed chain would work fine.