For the record, I am a practicing attorney licensed in NY State.
You can go from LLC Name (Foo, LLC) to Registered Agent Name in 30 seconds. You cannot do the same with Company Name to Bank Account (or from Bank Account to Owner Name).
I also use a registered agent for all my LLCs, but you don't have to. You can totally put your own address down when doing paperwork, and then your home address is open to the world.
EDIT: Elaborated
No, you can just register as a foreign LLC [1]. Notably, you do not have to disclose ownership to do business as a foreign, i.e. non-Californian, entity in California.
[1] http://bpd.cdn.sos.ca.gov/llc/forms/llc-5.pdf
Disclaimer: I am not a lawyer. This is not legal advice.
A state business registry does not. Which is why you use / might want a registered agent for your business. But you don't need to do the same thing at the bank (and in fact, I don't think most banks will allow an account without a name and a SSN to tie it back to)