Does a Subcontract need to be executed for a project located in CA, in order to submit a Stop Notice/bond claim?
No, a subcontract does not need to be in writing in order for stop notice rights or payment bond claim rights to be available. Though, proof that the work was done and proof of the value of the claim will likely be necessary (which can be shown via purchase orders, invoices, informal agreements, communications, etc.). For additional information on CA bond claims and stop notices, this page should be useful: California Bond Claim Guide and FAQs.
If you're facing a payment issue right now, these resources will be useful: (1) Send a Notice of Intent Before Making a Construction Bond Claim; and (2) How to Make a California Payment Bond Claim.
Not necessarily, as long as you can prove the reasonable value of labor/materials that were incorporated into the project.