I think it should be easy enough to feel a slipping pawl. Flip the bike upside-down on carpeting/grass, and pedal it slowly by hand. Or, remove the rear wheel and feel what the freewheel is like.
As mentioned above, that freewheel looks almost new.
A skipping chain can also be caused by a tight chain link. Flip the bike upside-down and pedal backwards slowly. Watch for the chain to hop as it travels through the derailleur. Or, slowly run the chain through your fingers, and make sure every link is loose. If you find a tight link, wiggle it back and forth until it loosens, or use the shelf on your chain tool to push the pin back slightly. Have you ever lubed the chain?
Slack in the chain can be a bit disconcerting. Usually not on every pedal stroke, but after coasting, or trying to back pedal. I suppose a sticking freewheel could cause slack instead of coasting. Also backpedaling when cross-chained, or slightly out of gear.