Please tell her not to worry - it will get better. Lot of babies fall asleep nursing. Try to prevent this because it's a terrible (and exhausting) habit for them to get into. Not letting the baby get to comfortable is the key and several moms gave great suggestions about how to keep them awake. I had to do those things until my daughters got used to staying awake. Also, her milk may be coming down fast. My girls would seriously nurse SUPER FAST, but it's because they were getting plenty in such a short time (7 mins on each, if I remember correctly). Even when my girls were older 3 & 4 months, they never nursed more than 22 minutes total. Every body is different, and she'll find her niche.
Being engorged is normal, at least it is for everyone I know at this early stage. Once her milk supply levels off, it will be fine. I know she's sore now, but keep pumping to relieve the pressure... but not too much that the supply increases. Le Leche League is awesome, free, and what every nursing moms needs when she's stressed about nursing.
As far as the vomiting, check with your dr, but as long as the baby is not loosing too much weight and is having enough wet & poopy diapers he is probably fine. My youngest would projectile vomit pretty frequently, but was gaining weight perfectly. She had a little reflux and the doctor precribed some meds for it, but it got better on it's own. They even did a test to make sure she didn't have a blockage in her intestine, but she was perfectly fine.
For gas, Mylicon (or the store brand alternatives) are a God send. My babies were both breastfed and I relied heavily on that stuff. Seriously, it's perfectly safe and it really did the trick with my girls and their gas.
Please tell your daughter not to worry or stress out about it. It takes a little time to get the hang of it for some moms, and if she's stressing out about it then figuring it out will just take longer. I am no expert by any means, but if she would like to talk things out or needs more reassurance please feel free to message me. Nursing is hard work (contrary to what many think), and by the time she gets it all down perfectly it will be time for him to move on to regular milk.
Best of Luck!