I am not an employer. I am an independent contractor without a doubt. Can you please answer the questions?
I am not aware of a California statute that obligates you to have a written contract. But without a written contract you are essentially entering into an oral agreement for your services.
If there is a breach of an oral agreement, there is a two year statute of limitations to file suit.
A breach of a written agreement has a 4 year statute to sue.
Oral agreements are harder to enforce in court, because the terms and conditions are not set out in written form and therefore more difficult to prove .
Hope that answers your question.