No, you paid for your unemployment compensation while you where working or rather your employer did. You do not have to pay it back and it will have no adverse affect on your credit history or when you apply for any positions in the future.
This is a common question. The answer is to look at your Marital Settlement Agreement and the Findings of Fact and Conclusions of law. It will say in those documents. If it does not say you have to pay for half of those kinds of things or that he has to ask your permission first, then you do not have to pay.
If you incorporate, such as a single-member LLC, you can avoid personal liability if anything goes wrong. You are free to contract with vendors, but if you do so and things go wrong you will be personally liable to them and, perhaps, any people that might be injured. Speak to a business attorney. It may cost a little money but if something goes wrong it could cost you a whole lot more. Good luck!
I dealt with this same situation last year and we were able to stop the counseling before it began. She probably is in contempt for (1) not giving you notice and (2) taking them to counseling without your permission. Find a local family law attorney and discuss this matter with them. Don't wait or things will only get worse.