For Bugsy, you could try playing some music minutes before you actually open the cage door so it distracts her from the creaking sound. The reason why she freaks out is probably because the sound is too sudden. That said, it probably is more practical in the long term if you could change the design altogether so that she can come out of the door herself rather than being picked up.
I’m not sure how Bandit’s true personality is like, but some rabbits just don’t like being petted in general. He could be one of them. Since you mentioned they are all rescues, it could also be that he was mistreated at his previous home and is hence a bit more wary towards oncoming hands. You didn’t mention how long he’s been with you, but you could try just sitting with him and refrain from touching him; instead let him come and sniff you instead until he’s used to you being around, or hand feed him some of his favourite treats so he recognises your scent. As for body language, you can look at his eyes and his body – if his eyes are extremely wide or his body is hunched, then he is most probably scared.
For Cooki, a month with you isn’t a very long time – I’ve read on here that some buns take months and even years to warm up to someone depending on their personality + experience with humans. If she flops and cleans in front of you, I’d say it’s good progress. Buns generally hate being picked up, so it’s not surprising that she squirms. Try not to pick her up unless absolutely necessary, otherwise you may also lose her trust before even gaining it. For nail clipping sessions, you can try clipping her nails when she is lounging or eating – basically when she’s distracted. You can just clip one nail at a time over a few days. Try putting the hay in her litter tray instead as bunnies like to eat as they poop. Do you have any chew toys for her to play with? That may divert her from chewing the wood boards.
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