Yes, it can be done, but you'll need to buy an Audi 016 Trans as well and go from there. Not worth it really, I've done it though. Seach threads by me and you'll find all you ever didn't want to know about getting a 016/944 hybrid to work.
Regarding LSD into 016. I used a Quaife LSD in my 016 and laterly in my 944S2 ring gear. Fits both. Works very well. But you have to strip your trans to install it. Don't know about old 944 clutch type LSD, but it should fit as the Quaife is designed to replace it. Why bother when you can get a new Quaife that won't break and does not need any maintenance, rebuilding etc.