If a police officer doesn't give you some form of documentation, and you don't receive a ticket in the mail, can you still be charged for a crime?

You ought to check with the court.  It may be that you can put yourself on calendar.  You should do this immediately to avoid being arrested on the warrant.  If you did not receive notice the warrant should be recalled without a finding of contempt.  If you are charged with an infraction, while a lawyer will be helpful it is not absolutely necessary it is more important to get yourself put on calendar and to have the warrant recalled.  The mere fact that you did not recieve notice, however does not mean, necessarily, that the case will be dismissed.  You may want to get the case put on calendar, have the warrant dismissed and then ask for time to get a lawyer.  Your request should be granted.      

Although I do not practice law in the State of California, here are my initial impressions.  There is clearly something missing from this story.  If your case made it onto the court's docket, there must have been some documentation demonstrating that you received an arrest, a ticket, or a summons.  In any event, at this point in time, you will need to hire an attorney and arrange to turn yourself in on the warrant.  If the cop truly made a mistake and you were really "let you go", then your attorney will be able to use that to argue that the warrant should be vacated and the charge(s) dismissed.  However, the first step in resolving this matter will be to hire an attorney to get to the bottom of this situation.  Good luck.