I'm in with everyone else. Just fix the parts that are broken and coming apart and put on a new binding. I'd, also, suggest you put a label on the back with the full story of the quilt, your name and the year you did it. That will mean so much to someone who has it in the future. These older projects are "very" special to us quilters.
Someone suggested you put on a new backing. Is the backing bad? If not, I wouldn't do it.
I have no idea about your thread question. A lot of people quilt with polyester thread because it is so durable. I've heard the rumors about it cutting into the fabric but have never seen it happen. I would think using polyester, although the cottons these days are very nice.
Great job on a first quilt.