Some public university libraries can even be free, e.g. the University of California system allows free in-library use of materials and computers to visitors (you just walk in, and if you want to use electronic resources, login as guest).
Yes and no. At Berkeley, frinstance, you won't be able to get into the main part of the main library without a university ID card of one form or another. (And the other parts of the main library often have a few homeless people sleeping in armchairs.) Still, you can always get into the Physics library and that'll have all the journals you need.
If you're a genuine citizen scientist (as opposed to a crackpot) there are always ways around it though. Talk to the librarians, they might be able to hook you up with a card in one way or another.
Interesting. UC Santa Cruz has no card checking at all, and pretty explicitly allows the general public to visit. You can even use the WiFi without any sort of account. It's a little out of the way, though.