Yes, it is only the 83's that have the frame problem. Mine broke @92Kmiles, and I purchased an 84 frame to replace (BTW, this is a big, time consuming job!!). The 84 frame was re-designed where the mono-shock and the center stand attach. It was made of heavier gauge steel, and had a couple of gussets added. It seems that Yammie must have discovered this in testing, but not in time for the 83's.
The 2nd gear issue is for all of the 83, 84 and part of 85 production. I don't know where I put the service bulletin that tells what S/N is the dividing point in 85 production, but I'm sure you can find it on this site somewhere if you are considering an 85.
The good news......the 83-85's are 50# lighter and the 1200's have hotter camshafts, so top end power is higher. If you wind them up, the 1200's are faster than 1300's. However, you have to shift a bit more on the 1200 bikes.