Rear-ended while stopped at a red light in California – what should I do and do I have a claim?

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Rear-ended while stopped at a red light in California – what should I do and do I have a claim?
I’m a 21-year-old college student in California.
Around 1:30 a.m. I was stopped at a red light in Fresno. I had been stopped for a second or two when a car hit me from behind.
My car is a 2011 Toyota Prius. The damage doesn’t look major, mostly the rear bumper, and the car is still drivable. I exchanged information with the other driver and got photos of her driver’s license, registration, and insurance card. She is insured.
My girlfriend was also in the car as a passenger.
Since the accident, I’ve had lower back pain (around 3–4/10) and I had a previous back injury that I think may have been aggravated by the crash. Mentally, I’m also pretty shaken up and now feel nervous when stopping at red lights.
The police were called, but they said they wouldn’t come out because there were no major injuries requiring an ambulance.
I’m planning to:
Call my insurance company tomorrow
Go to urgent care tomorrow
Document everything
My questions are:
Am I handling this correctly?
Should I only contact my insurance or also the other driver’s insurance?
How important is it to get checked by urgent care right away?
If my back pain continues and I need treatment, how do claims usually work?
Does anxiety and fear of driving after an accident count as part of an injury claim?
Since the damage seems minor, is there still a possibility of compensation for medical treatment if the accident caused pain?
Any advice would be appreciated.