Congratulations on the new position
Some of the pre-employment physicals are very simple brief affairs where they basically check your pulse to make sure that you are actually alive, make you pee in a cup and that's it. Others are 2-hour ordeals like the one my current company put me through with actual physical examination, lung function testing, EKG, blood tests, hearing/vision screening, hair/urine drug screening. Most companies fall somewhere in between based on my experiences.
Most will ask on the forms about prescription drugs that you are taking. The health information you give out does not go to everyone in the new company; generally only the HR person has access to that information and can't share it under law. You don't have to disclose to anyone at your new employer about your Bipolar and/or anxiety if you don't want to.
About drug testing: Most companies run 11 or 12 panel urine screens. The drugs that are being screened for a drugs of abuse and not prescription drugs. If you are taking normal prescription doses of the benzodiaziprine, then it should not be an issue. You don't have to write it down on the drug screening form. The only time that there is a "positive" for something like a benzo or other prescription drug is when someone is taking it to abuse it at much higher doses. I did have a friend with one exception to that but as long as you have your prescribing doctor's name and phone number then you are okay. The lab will call you if there is an issue but it doesn't sound like it will be.