Just fill out an application. It is either free or inexpensive for residents in the area served by the library. Non-residents usually have to pay a substantially higher fee.
Yes, you can take a picture anywhere, of anything, as long as you, yourself, are standing on public property. Here are the general rules for the United States:
"Local, state, and national laws may...
Definitely!where I live, we have 3 public libraries! (if you don't know the UK that well, I live in Sheffield, which is in lincolnshire *kind of near London*)