If I had an accident and it was the other driver's fault, will I have to pay for his repairs since I didn't have a license?

Under Texas law, whether or not you have a drivers license is irrelevant to who was at fault in the accident.  However, count on the other driver's insurance company to try to intimidate you into thinking that your lack of a drivers license makes you liable.  Just call the adjuster's bluff and focus only on the events surrounding the accident.