Will paid incorrect debts be refunded?

If you owed B money also then no. If you did not owe them money they would apply it to your next statement. If there is no future billing they should send it back but you might alert them to the error to be sure.

I paid Company A in full, so Company B was paid in error. Should I send a Debt Verification letter to both Company A & B?

I think you have to go into a little more detail. Did you actually owe company B or not? Your first post made it sound like your payment for A went to B but in your last post you said that both were paid? Is company B a collection agency for company A and did you pay the same debt twice? please explain...

Yes I paid both company A and company B. Company B was a collection agency for company A. I paid the same debt twice.