It is obvious that you don't know a whole lot about unemployment insurance. But you should immediately file a claim if you aren't working. You aren't required to know anything about it, there is no downside to filing a claim, even if that claim should be rejected. You aren't going to have to produce anything or do anything that you won't be told about when you get there. You aren't supposed to go in and argue brilliantly about your eligibility or go and prep up and then file. You will only be paid for the weeks after you have filed, regardless of how long you have been out of work. You will be required to produce some valid I.D., which may be an out of state driver's license as far as I've ever heard. Perhaps CA has changed something and will only accept valid in-state driver's licenses as a form of I.D. but I've not heard of it.
That said, you should quickly file a claim. You should file the claim in whatever state you now live and where you wish to do your worksearches. If you do not have adequate wages to set up a claim in this state, they will do what is called an interstate claim for you, bringing in the wages from the state(s) where you do have wages. But your state of residence is where the claim should be filed, regardless of what state it is for.
When you file the initial claim, the system will pull up a record they have of all the wages that you have had PAID IN FOR YOU by employers in this state, not taxes you've paid in for yourself anywhere. Your unemployment entitlement will be based on this state's maximums and minimums, using the standards they apply to set up claims. This will be based on the first four of the last five completed quarters of work, roughly the work you have done for covered employers in the last 18 months. You will be responsible for telling them about work you have done for out of state employers, and they will send for and pull in any wages that have been paid to you from covered employers in this state.
After they've determined if you have enough covered wages to set up a claim, they'll then determine whether you are eligible based on the reason for leaving your last place of work. If you are determined out of work through no fault of your own, whether you have been laid off or have been terminated for cause and the cause is determined not to be disqualifying, or possibly if you have a good work related reason to quit your job, you can be approved.
Then the system starts determining whether or not you are ready to make a job search, be able and available and actively seeking work. This is where having a valid driver's license comes in. If you tell the system you can't work right now because your car isn't running, or because you can't afford to get a CA driver's license and your old one has expired, or if you are normally a truck driver and you don't have the appropriate CDL license to perform your job, this may be a problem. But having a valid out of state driver's license isn't a problem that I've ever seen. It could very easily be fixed, all you have to do is get one, if they were to tell you you had to have it for some reason. If you didn't have enough valid I.D. to get an I-9 filled out for you by an employer, you wouldn't have enough to file for unemployment benefits anyway, but in any case, don't sit around here asking moot questions about it, file your claim and let them tell you what you need to get or have.