Its sounds like everything she is doing is very typical for 2 year olds. Its essential that you accept her behavior as normal and avoid power struggles.
Really, the key to the next 2 years is avoiding power struggles.
Bathroom - she's probably not going to let you use the bathroom by yourself. She'll want to come in with you.
Cleaning - I'm assuming you need to clean and she doesn't want you to, right? And not that you told her to clean up and she refused. Its okay to prioritize her needs for companionship and play over cleaning during this time, and let the house be a little dirtier than usual. It can also help to get her involved even though that usually means more mess and time for you, it makes her feel connected to you and not in opposition to you. Pull up a chair to the sink for her and let her splash, make a game out of throwing the clothes in the washing machine, let her pour the soap in and close the lid, etc.
Going to sleep is hard. At that age, I went to bed with DD and nursed her to sleep. Actually, at 4.5, I still go to bed with DD. After she falls asleep, I can either get up if I'm not sleepy or usually I get up in the morning before she does to get some cleaning done or just have my own time.
Nursing... I'm guessing you mean the acrobatics associated with toddlers nursing? The uncomfortable twisting and turning? Its possible, without getting angry or upset, to encourage her to lay down on your lap to nurse. Gentle persistence will pay off. If you're talking about the frequency of nursing, I'd just stop what I'm doing and nurse her as often as she wants. Avoiding power struggles. Nursing frequently is sometimes about the need to repair a disconnect and attach to you more. Actually toddlers usually just love to nurse.
Food just shouldn't be an issue. Either put those things that she takes one bite of then wants a new one out of site, leaving only one in view, or package the unused portion up for later (bananas can be frozen for use in cooking). One thing many mamas do with their toddlers and young children is put out a snack tray with lots of things to choose from (pieces of cheese, carrot sticks, veggie booty, a few slices of apple, etc) and the child can graze. Another thing that works really well is having her own spot in the frig that she can access anytime that has snacks just for her (ours has Diego yogurt drinks, little baggies of a few black olives, little containers of applesauce, string cheese, half an apple in a baggie, small pieces of nitrate-free lunch meats, and fruit leather).
Avoiding power struggles is what makes this time of her life so much easier on you, and makes your attachment positive and fun. Use humor to redirect rather than direct opposition, get silly, spend lots of time on the floor together. The hitting and tantrums should decrease as well (although not completely. Volatility is characteristic of the toddler, as well).
