Does bodyweight really matter that much?
Yes, more than for almost any other lift. Every kilo you're carrying is a kilo you lift on every rep. It's not a moral issue, it's arithmetic, and it's why two people doing identical training can be months apart.
Can I just use the assisted pull-up machine at the gym?
It works, but it stabilises you in a way a bar doesn't. Use it as a supplement, not a replacement for hangs and negatives.
My elbows ache. Is that normal?
Muscle soreness in your lats and forearms is expected. Ache on the inside of the elbow that lingers into the next day usually means too many negatives too fast. Drop to two sets and see if it settles.
Chin-ups or pull-ups?
Chin-ups (palms toward you) are easier because your biceps help. If you want the first rep sooner, start there. Nobody serious will hold it against you.