Besides the earliest extant version of a ballad that has long been in print only tells us the ballad originated 'some time' before that printing. Some have suggested 17th century. I'd go somewhere in between and say second quarter of 18thc, but these are wild guesses. I think if it had happened say c1770 there would be records and even newspaper reports.

Regarding Moore, there are several art songs called 'Molly Bawn' and one of these is more likely to have had an influence on Moore.

Uncle/father. You've really got me confused here, Richie. Why shouldn't it be her Uncle, probably her guardian or a very influential member of her family? He may have been the leading figure in prosecuting Jamie.

It would seem his father was very confident in his story being believed, but until we can find some proper background info we are fishing in the dark.