The right shifter for the 7 speed rear cog has had some issues. It shifts ok but when I move the large lever to move to a bigger cog wheel, I have to pull it back manually so something with the spring load isn't functioning properly.
I will post a picture of the chain rings...the chewed up teeth might actually be designed like that so I will let you guys use your internet armchair expertise to let me know if it is in need of replacement or not. They aren't sharp jaws teeth by any means but several of them are ground to about half the height of the others.
Not unheard of for early Rapidfire shifters to be mis-behaving. Unfortunately the early models are quiet complex to take apart and if the internals are worn there's not a lot you can do. Look for a good used set or switch to more-reliable thumbshifters
Yes some chainrings have shallow teeth to aid shifting. More important is the gaps between the teeth that the chain rests against... If the teeth start looking like a shark's fin then you might need a new chain/cassette too, depending on how long the bike has been run with a stretched chain.