I don't understand why there was a question mark after ね in the first message. Are they asking for my agreement that I think I'm good at Japanese?

It is like them saying, "You're good at Japanese, aren't you?"

You're supposed to politely decline.

I don't know about you, but I was taught growing up that denying someone else's compliment was rude, and that you should always gracefully say "Thank you," even if you disagree with the compliment. The opposite is true in Japanese. Even if you do think you're hot stuff, it is better to politely decline. "You're good at cooking!", "No, I just followed the recipe." That sort of thing.

Secondly, why is there a の after 住んでる? I've seen this before in "何をしているの". I also read on StackExchange that it's only used in casual contexts.

It is a question marker like か but used with non-"desu/masu" sentences. It isn't necessary, as you can accomplish the same thing by using a question intonation and just saying, "どこに住んでる？" but using の is very common.