I won a sum of money from the California State Lottery, I don't have an ID card from the state of California. I have an out-of-state Washington paper ID and a birth certificate. Can I cash my check from Lottery with the identification I have or is there any other way to get my money?
If you don't have a bank account then open one and use the check as your opening deposit. You'll need to obtain a Social Security card as well (you didn't indicate you have one or not, so I'll assume you don't), but they're free and quick to apply for.
I assume you're from Washington state, so there should be no problem with you opening an account in Washington, no matter where the check is from. Because of the amount of the check, the bank will probably only let yo have a part of the amount for a few days until it clears, but federal banking rules require quick clearance of out-of-state checks (I think it's 48 hours now), so you'd have access to at least half of the money right away and the rest in a few days.