Will gaining back weight help my sagging breasts?

Depends. Many women who lose large amounts of weight will complain there breasts now sag. Looking back they probably sagged at the heavy weight but were fuller. Large, un-augmented breasts usually to sag. Gaining back the weight may add fullness but is not likely to lift the breasts.
No. If they are saggy they'll just be bigger and saggy.
No. Once breast become ptotic or "saggy" from weight loss they generally do not improve with weight gain. They just become bigger and "saggy" there are procedures such as mastopexy that can help. You need to consult with a board certified plastic surgeon.
Fuller not lifted. To some degree this may make your breasts appear fuller and not as flat or deflated appearing (involutional atrophy). However it will not lift the breasts. This can only be achieved with a breast lift or bra.