I have a road bike, 10 speed 11-25t cassette, and I just have a question in regards to replacing the smallest cog.

I am attempting to replace the 11t cog on my shimano 10 speed cassette 11-25t, because when I'm on highest or smallest cog it skips. It makes that sensation or it feels like your shifting from the 9th to the 10th gear(from the 2nd smallest to the smallest cog) every so often, like the chain is getting caught on one of the teeth and lifts up and then lets it go.

I got my bike just recently from a guy who was selling it, I had it tuned up from a local bike shop and they replaced the chain they said that it stretched already and needs replacing. Could that have been the problem, a different chain? The chain that they replaced it with came off from one of the 10 speed bike, that they said was a take off and was hardly used.

should I replace the chain with a brand new one or should I replace the 11t or the 12t cog or both? And if I do replace the 11t or 12t cog, can I replace it with any 11t or 12t cog that I see on e-bay, whether it be for a mountain or road, shimano or not? Or is there a specific 11t and 12t cog for the shimano 105 10 speed cassette 11-25t? If there are specific 11t and 12t cog for the shimano 105 10 speed cassette 11-25t, then can anyone tell me which specific ones I should buy or which ones are compatible for it?