I tried to find a list of states with costs for obtaining copies of birth certificate but couldn't find one but judging from the google hits I think you are right in that there is a charge in all states for them (cost will definitely vary though).
Couldn't find any information on what someone could do if they didn't know where they were born- don't know how common that is- would think at most they'd know what state they were born in. In this day and age it may be easy enough to find using their name and their parents' names... err if known. Fascinating... now I'm curious lol.
