Are you sure? The answer to question 2 in the opening post, seems to be that the BRP is what is needed to prove a right to work in the UK and to access the NHS without being billed? The US passport only has the 30 day entry vignette.
This is from the government site-
What’s on your BRP
Your BRP will include:
your name, date and place of birth
your fingerprints and a photo of your face (this is your biometric information)
your immigration status and any conditions of your stay
whether you can access public funds, for example benefits and health services
You may have a National Insurance (NI) number printed on the back of your BRP. Not all BRPs have this - it depends on factors like the date it was issued and your visa status.
Why you need a BRP
You can use your BRP to confirm your:
identity
right to study or work in the UK
right to any public services or benefits you’re entitled to
https://www.gov.uk/biometric-residence-permits
And this-
You should bring your biometric residence permit with you when you access healthcare in the UK.
https://www.gov.uk/healthcare-immigration-application/overview
When I signed up with my GP they required my passport, BRP card and proof of my address when I joined the centre. They made copies of all of them.
When I was working with an employment agency when looking for work I was required to bring my passport, BRP card and my NINo to prove my right to work in the UK on my initial visit with them. If you work with more than one agency, each will require the same thing.
The three employers I have been hired by also required all of those things at time of hiring. Wouldn't have been hired without it (it is the passport). (Current employer requires passport of their UK employees).
I had to have my passport to apply for my NINo.
You must have your passport to apply for your UK provisional driving permit of which you can't take any part of the driving test (theory included) without it.
Yes, you could use your BRP card as an ID, but why would you?! It costs about £300 to replace and you'll be without your passport for several weeks. It's your visa, why would you carry it around? It's as important, if not more important, than my US passport!
Get your UK provisional license and use that as your ID over here. It's cheap to get, under £40. But you will be without your passport for a week or two depending on how long it takes to process. It's not too long.
Eta: stupid phone keyboard had to fix some weird words
Oh, should also add that if you are looking at a job through an agency and you've never worked with them before they require you to meet with them first before you could have an interview with the actual employer. At which they will require your passport, BRP and NINo for their records.