Although this looks like a request for specific legal advice, it can be treated as a general-interest question about legal obligations and risks (not practical advice).
When you buy a car, you ordinarily give the seller money (cash is fine, and might be the only thing they will accept), and they hand yo the car and the "pink slip" i.e. the title – which must be filled out to indicate that they have transferred title to you. You then have to do everything else. The seller does not normally go the the DMV with you, but you have to be sure that the title is properly transferred. You don't need an SSN to actually buy property, but that doesn't mean that you won't need one for a later purpose related to the car. Writing a false sale amount is a legal red flag. Liability insurance is mandatory, so you should have the vehicle insured per the state rule before you drive it. That is the summary. Now the details.
In California, the seller must notify the state of the transfer within 5 days. The buyer must do their part within 10 days. For the buyer, this means providing the original title, bill of sale if the seller is not the person on the title, smog certificate (perhaps) and odometer reading (perhaps). There are also fees. This page gives you the relevant forms. A notary may be required for some forms (meaning the owner/seller has to go somewhere), for example to apply for a duplicate title (if they lost it)
This page describes the liability requirements in California. You must be insured if you drive or park on California roadways, which implies that you have to get insurance before or at the time you buy the car. Under CVC 1653.5
(a) Each form prescribed by the department for use by an applicant for
the issuance or renewal by the department of a driver’s license or
identification card pursuant to Division 6 (commencing with Section
12500) shall contain a section for the applicant’s social security
account number.
So you cannot drive (get a driver's license) without an SSN.
