1. Yes, that is perfectly fine.
2. Yes and you may find she can be trusted with more freedom once she’s older.
3. It will, but she will trust you more if you don’t pick her up all the time (just when it’s necessary). What type of cage does she have? Can she get in and out on her own? How about making a step for her so she can get up on the couch on her own. My buns would use the dog ramp to get on the couch all the time.
4. If you are the bringer of food and treats, I don’t think you’ll have a problem with her loving you. Bunnies show their affection in different ways though, especially as they grow into adulthood. Don’t think that she doesn’t love you if she suddenly decides cuddling isn’t the bees knees anymore. She may find other ways to show you affection like licking, nose bonking, seeking out pets, flopping near you, etc. Each bunny is different.
5. Is her nose going under water? You don’t want her to accidentally inhale the water. Some bunnies do have strange drinking happens, I have noticed. Leopold does this weird thing where he gathers water in his mouth and then spreads it over his chest and shoulders, like he’s giving himself a bath. Weirdo.