The skipping has been addressed by others. I would think the new freewheel will address that, along with a new chain.
As for the gearing, I’m pretty sure the OP has a true 10 speed bike, as in a 5 speed freewheel.
I believe that a 13t is the smallest you can get, I know that there were a few 12t freewheels, but I thought they came along later in the 80’s.
I don’t see an issue with going to a 52t (52/40) and the 13-22 freewheel. Clean the housings and use new cables, adjust the FD and you should be set.
John
Edit added: If you are getting a 30+ year old NOS 5 speed Suntour freewheel, you should probably lube it first. It has been too many years but I used to periodically remove the cogs, soak the freewheel body, let it dry and drip oil through it.
I would guess that someone familiar with your winter weather can expand on this.
Last edited by 70sSanO; 09-17-22 at 11:31 AM.