Are these sentences correct?

Usually, people get picked on, not inabilities. I would reword the first sentence to something like Tanya picked on Sian because she couldn’t swim.

Your other sentence is fine.

Yes.

Thanks. I thought the same about people and not inabilities getting picked on; however, a quick Google search revealed that people have said something along the lines of "picking on her/his inability to", so I got confused.

The confusion there might be because Sian could say, "Tanya picked on me not being able to swim." Although the correct form would by to use "my" instead of "me", most Americans would say it as I have typed it and almost no one would say "my inability" unless they wanted to be picked on for being a nerd.