Am I obligated to pay for work that was done if I found out after that the company was unlicensed?

I do not know the answer to your question with certainty. Generally, in GA, you can not use something like that as a defense, unless it was the proximate cause of actual damages that you suffered, as a result of their lack of license. You would not have a fraud claim, unless they represented that they were licensed, knew you would rely on that mis-representation, you in fact did rely on that representation, and suffered damages because of that misrepresentation. However, one could argue that you failed to exercise due diligence yourself.

I would think yes, their problems are theirs with the regulatory authorities. That you got what you wanted and are satisfied with the work should tell you the regulation (and additional cost it would have visited upon you) was unnecessary. That you got the benefit for your bargain creates a moral, and most probably legal, obligation to pay for it.

Most likely yes.

Morally, yes.