You've got a number of issues here. First is that 7-speed cassette has the cogs spaced more narrowly than a 6-speed freewheel or cassette, and 6-speed Shimano shifters are designed for a 6-speed freewheel or cassette. Which means that if you line the derailler's increments up so that the chain is perfectly in-line while on the middle cog of the cassette, it won't be perfectly lined up as it gets toward the largest and smallest cogs. This may cause the chain skipping, although it would make more sense if your chain skipped on the larger or smaller cogs but not in the middle.
(I assume you're using a spacer to mount the 7-speed cassette on an 8-speed freehub body. Otherwise the cassette can't be fastened down tightely and could slide from side-to-side.)
Current chains designed for 5/6/7/8 speeds are fine for your situation. Back in the 80's there were "narrow" (for that era) chains designed for 6-speed sprockets. These would be slightly too wide for a 7-speed cassette where the cogs are more closely spaced.
Another possibility is that the chain and/or cassette are worn-down. Teeth get shaped like shark-fin profile when worn down, and the chain starts to skip. Get a chain checker (I trust this a lot more than measuring with a ruler) and see if the chain is worn. If so, you may need to replace it, and will probably need to replace your cogs if the chain is worn. If you're going to replace cogs and chain, you may as well get 8 (or 9 or 10) speed cassette and shifters to match. They'll fit on the hub that you're currently using.