The wobble typically causes no problems. There's an outside chance it may be exacerbating the problem, but it isn't the root of it.

Skipping on the small cogs is the classic symptom of a worn out freewheel. BITD they were ridden until they started to skip, and then replaced. Typically you would go through two chains per freewheel, so this might be about 10-15 thousand miles. Shimano freewheels weren't long lasting, so I'd guess the lower end. Chain cleanliness makes a difference too. At any rate this is hardly surprising as mostly racers used things like Dura Ace freewheels when 6 speed was standard, and this one has likely seen many thousands of miles. The wear isn't usually visually observable. You can tell because the chain starts to skip.

AFA remedies: If possible, replace the offending cogs. That's what used to be done if the cogs were available and money was short. Probably hard to find now. Definitely check your hanger alignment, as a bent hanger can cause the same symptoms. Also check for stiff links in the chain. The problem might be minimalized by tweaking your axle position in the dropouts, if you have horizontal dropouts. Move the hub all the way back. Try adding a link or removing a link from the chain, or try a different chain.