As Phil says I would take it back to the shop and ask them to adjust the limit screws on the derailleur. Those are meant to stop the chain from shifting entirely off the freewheel. This should be one of the basic things they do as part of any tune up. Giving the bike a decent cleaning is also generally what's included in a basic tune up. I'm surprised whoever completed the "tune-up" didn't degrease your freewheel cogs a little bit, and at least get the built-up gunk off your derailleur pulleys. I'd be skeptical they touched the drivetrain at all if that's what your derailleur, chain, and freewheel came away looking like in the second picture. I'd take it back and show them the picture. As a former service mechanic I'd call that unacceptable.
Regarding the "short" teeth on the big cog there - that is normal and there by design. The theory was it helps the chain engage the large cog when shifting into it, normally a somewhat difficult shift due to the size of the last cog.
Lastly - nice job cleaning that area up. Just be wary with using a lot of degreaser around the freewheel - there are a lot of bearings inside the freewheel and inside the rear hub that need grease, and spraying that area down with a ton of degreaser or soapy water or solvent might dissolve some of the grease. I'll generally take the freewheel off the hub if I want to deep clean it.