Got some small parts in to properly mount the wheel. Most notably the bolts for the axle. Turns out the 5-speed uses a different sized bolt that the older 3-speeds.
I wanted to mount the bottom bracket and put the chaincase back on but it gets stuck.
The first couple of threads go just fine but then after about 4mm in the whole assembly goes sideways and gets stuck. I can't see any crossthreading though. The old bottom bracket that came out has the exact same problem so it seems like the problem is the frame. I'll probably drop by a bike shop tomorrow and ask if they can retap the threads.
To be continued!