Why is my chain riding up and skipping teeth on the freewheel?
Need better pictures to get a clear idea as to what is going on here, but assuming that the chain tension is good and the derailleur spring is okay, I'd check to make sure your derailleur hanger is straight. Especially on older derailleurs, this can allow the chain to be received by the cogs at an angle whereby the cog above and/or cog below can 'trap' the chain.

Checking the hanger is easiest with the right tool, it's a bit pricey ($60 - search on "Park Tool Derailleur Hanger Alignment Gauge - DAG-2") so best bet would be to hit your local bike shop and have them lend a hand.