1. You should be able to remount a sling stud with a nut on it a little farther forward, the forend is hollow, but mine are already just 1½" from the front end, that should be plenty of room.
2. Can you see the rear sight over the scope rail? I can't unless the rear sight is all the way to the top of its travel.
3 &4. The FAQs has the skinny on the Weaver 82, it's a poor mounting choice. With a little trimming of the hammer spur, you can use medium rings.
5. Fill the stock with shot, either poured in, or in a sock, to add weight. If you shorten the stock, the recoil pad won't fit anymore since the stock profile is tapered, the shorter it is, the smaller the butt, the pad will be too big. I'd get a youth syn stock for it.
Tim
1½-4½x32 Banner in medium rings on H&R rail