Hi S.! My now 11 month old DD refused the baby food I made for her as well, so she has been on solids since about 6 mos. And she didn't get her first tooth until she was 9.5 mos old, so even without teeth, they can gum up food pretty well. Some things I fed her at that age were avocado, banana pieces, any veggie steamed soft, peaches, mango, tofu, small bites of chicken, turkey, ground beef, mac n cheese (messy, but they love it), plain pasta, toast with butter or jam, pancakes, waffles, cheerios, graham crackers, hot dogs, rice, hummos, blueberries. I have a 2 year old DS, and I feed my DD anything I feed him, I just make the bites smaller. You'd be amazed what they can/will eat at this age, and if you start them early they are more likely to accept more foods when they hit the ultra-picky toddler stage.
As for the sleeping issue, I am willing to bet she doesn't need that 2am feeding. So long as she is following the growth curve, she shouldn't need to eat overnight. Unless you are BFing (which I did not), in which case I understand BFing babies may eat at night still at this age. At any rate, in my opinion I wouldn't let my baby cry until they vomit, either. I don't think it accomplishes anything. She needs to learn how to self-soothe. One of my favorite sleep books is Dr. Weisbluth's Healthy Sleep Habits, Happy Child. He has several methods of "sleep training" that will vary depending on your child's needs. I also liked the Baby Whisperer book that another PP recommended. Read a few sleep books -- every baby is different, and your baby will respond to one theory over another.
Not all babies learn to sleep through the night on the same scehdule. Hang in there, and it will happen when she is ready. You're doing great!