When filing a counterclaim, are the names of the defendant and plaintiff switched at the top of the pleading?

The style (name) of the case remains the same. If you are a Defendant andy you are filing an answer and counterclaim, you would refer to it as "Defendant/Counter-Plaintiff's Original [or Amended or whatever] Answer and Counterclaim." That is one way to do it. There really is no magic to as long as you make clear what you are asserting. But the style and positions of the parties do not change from what was originally named. The only variance of that is that in the style after "Plaintiff" you can add /Counter-Defendant and after "Defendant" you can add "Counter-Plaintiff". Hope that helps.

No. The style never changes. All of the pleadings should have the same style, cause number and court on them. It sounds like you are a pro se litigant. If so, I strongly recommend that you promptly consult a lawyer. Without one, you will almost certainly lose, especially if your adversary has a lawyer. Good luck.