Are you sure Beemer is a girl? Female Amazons tend to get hormonal for a few days (and not generally aggressive) whereas males will get that way for a few months (and be pretty aggressive), so I wonder, if Beemer really is a girl, if this is hormonal or if she's just trying to communicate something to you.
If you're there to supervise, have parrot proofed, and don't live with any predators I don't see any reason why Beemer couldn't get on the floor and wander around. The birds at the rescue are free to do this 24/7. Kazi is not since we have dogs and cats. Does she have a training stand or play gym she can be on when not in her cage?
Does Beemer like her cage? If so she may be 'training' you to return her there by biting. Don't let her do this. You don't want her to learn that the way to get what she wants is by biting. Kazi was taught this at one point in time and we're having to deal with that now. He still bites when he wants something I'm not giving him fast enough. What it actually gets him is the opposite. So when Kazi bites me because he doesn't want to go into his cage when I leave to go grocery shopping, he goes in. When he bites me because I don't give him his almond fast enough, it gets fed to a dog in front of him. He's learning and has nearly stopped biting me entirely to get his way. For a couple of days I had to use a stick to get him up when he was feeling very strongly about getting his way, but persistence has paid off, so don't give up!
As to the screaming, she probably stops when you put her in the unfamiliar room because parrots are prey and they're not about to alert any predators to their existence. That being an unfamiliar room, Beemer doesn't know what predators may be lurking, so she's quiet, if you keep putting her in there eventually she'll realize it's safe and start screaming there too. She's likely screaming in her cage because she wants out or otherwise wants your attention. Do you do any face to face training with Beemer? That alone has helped my relationship with Kazi grow by leaps and bounds. Kazi and I spend about ten minutes a day working on our tricks. Kazi loves these training sessions and so do I. He gets to do something stimulating and fun and we end up working on communicating with one another at the same time. Plus the commands go a long way toward presenting a distraction should I need one. I just have to show Kazi an almond sliver and and ask, "Who wants an almond?" and no matter what he's doing he 'raises his hand' and gets the almond. He's distracted from whatever he was doing and it's all done on a positive note, so the bond is also strengthened.
I hope this helps!
