Why don't you try the C&V subforum here at bikeforums?
I think the chain is fine.
Check the major bearings for wobble (when not spinning-like if the wheel was loose on the axle). Those would be the hubs, the bottom bracket, and the headset. It is best to take them down and regrease because old grease can dry up. If they don't wobble and you really don't want to buy a large crescent wrench, hook spanner, and cone wrenches, then at a bare minimum oil them.
Having the wheels tensioned and trued would be a bit pricey but would go a long way toward making the wheelset last if you plan to ride the thing at all.
I rarely replace cables myself; they usually work fine.
If solvent won't free your brakes and you do wind up taking them apart, they go back together with a nut and a locknut on the pivot. the nut should not be cranked down onto the brake arms but should be held steady with a small, flat wrench while the outer locknut is tightened against it. Incorrect adjustment may be the problem.