It's slipping on the smallest sprocket because it has the least number of teeth engaged.
It is possible to wear out chainrings, but there's a rhythm to it. A cassette will usually outlast several chains, and a chainring will usually outlast several cassettes. The key is to replace the chain before it gets too bad.

Haven't tried adding a few fresh links to a chain in use, so I can't offer much help there. But I'd suspect stiff links first.

If you've ridden one chain to destruction, then it would probably have been a good idea to replace the cassette as well.

Looking at a cassette is a so-so method of checking for wear. With all the ramps and undercuts it's really hard to tell what is wear pattern and what is engineered.