My wife and I both enjoy this game a lot on our phones and we're thinking of buying it on PC so that we can play it together. Is it possible to buy it on my account, install it on both of our machines, and play together? Or do we need to each buy our own to play together?

Having already bought the game twice I'd rather not have to buy it a fourth time just to do co-op.

the game supports Lan so in theory it should work, but I never tested it so maybe someone else here can let you know just to be safe.

yes it is, that is how my wife and I play.

Yes, with windows/linux/mac players. You can use one install from the gog build (even the steam build just copy the folder contents from steam install) and then run using ip play (you can also remotely open up the ports the game uses to allow players on two seperate networks to play with out needing steam or gog client to get the game going). Also it does local lan scan so you don't even need to know the IP on the local lan.