1) babies this young, usually do fall asleep after nursing. Because they are full. They then drop off the nipple.
2) 3 months old is a Growth-Spurt time. So yes, feeding frequency increases as well as amounts taken in.
3) "Cluster Feeding" is, when a baby needs to feed even every.single.hour. And this is normal and indicates changing growth and growth spurts.
4) Intake in a baby, has to keep up with them. Not the other way around. Otherwise, they will NOT be getting enough intake. And feed on-demand.
This does not sound unusual to me at all.
Sleep in a baby, is NEVER static. It does change and per the baby's age stage and development and per hunger and per growth-spurts and per teething etc.
My kids, whenever hitting growth-spurts, woke a TON. I woke too and nursed on demand. They grew like weeds and had super GINORMOUS appetites.
But yes, as the other person said, make sure she is feeding adequately.... use both breasts per feeding. She needs to get to the hind-milk.
Also, not all babies will take a bottle. My daughter was like that. But my son would.
Your baby used to take a bottle. Now she won't. So you just need to nurse. My daughter would flat out refuse any type of bottle.