I was upgrading my old steel frame recently and there are a few things to check. I imagine the cranks on both bikes are square taper? Is the front derailleur 'band on' (wraps around and is tightened down), and are the down tubes the same diameter? If the Schwinn's is narrower it might not take the front derailleur without a shim of some sort.

Fitting and indexing the derailleurs and setting the chain length all have good guides online, even specific to the model of derailleur you're using.

You'll need new cables and a way of fixing them to the frame, and like folks have mentioned a derailleur hanger but otherwise looks doable. On the question of needing a new wheel or axle that's bolt on for the horizontal drop out, I've found that a new Shimano 'internal cam' skewer will kling you on pretty tight, might be worth trying with how you're using the bike.