The last time I checked and this is old since I live in CA in the 90's it does not matter what value you say you paid in a private sale. In CA they will verify what you claimed to pay against their database of car values. Remember CA has over 30M register vehicle and can easily determine FMV of a car and they will use this number not what you say unless you paid more.
Yes you can register as non-op and it is $19, http://www.dmv.ca.gov/vr/fees/miscellaneous.htm
I can not remember if you are still required to pay the registration tax on the vehicle. I do know that if you no-op you can not park the car on the street, actually it may not be even in your driveway visible, so it may need to be covered.