1. Are the rabbits spayed/neutered? Two rabbits that are not spayed/neutered have a very high chance of either fighting or mating.
2. If the answer to Question #1 is no, then the rabbits are most likely not bonded properly and hence will have to be separated to prevent potential fights. They will have to be spayed/neutered and then go through an intentional bonding process if you ever want them to be together again. Also, since you mentioned “she”, it’s important for female bunnies to be spayed to avoid potential health problems. Females have a high chance of getting uterine cancer if left unspayed. They are also notoriously territorial so hormonal behaviours will be quite difficult to deal with.
To answer your questions, yes most rabbits hate being picked up. This is because they are prey animals and being hoisted off the ground makes them feel like they are being caught by a “predator” (i.e. you). In situations where it’s absolutely necessary to pick them up though, try to make it quick (but careful) and not hold them for long periods of time unnecessarily (e.g. for cuddles etc).
She may be a bit wary of you after you pick her up. To gain back her trust, sit on the ground with her and don’t try to touch her. Let her sniff and explore you without doing anything, so that she doesn’t see you as someone out to harm her. You can also give her bunny safe treats (e.g. a small piece of banana, carrot or apple) so that she associates you with good stuff.
Since it’s a small door, it could be a better idea to get a bunny carrier and try to lure her into it using treats instead of picking her up. If you’re planning to have her in your room, make sure it’s fully bunny proof. Note that it’s difficult to let a bunny free roam if she’s not spayed because it’s highly likely she will pee and poop everywhere to mark territory.
You can refer to the BUNNY INFO section of this forum to learn more about bunnies, including diet, bonding, living space etc. It’s really helpful 