Everyone has a regional accent. Normally, you can't hear your own, but travel to another area and you will soon see that you don't sound the same as the people who live there. They will think you have an accent and you will think they have an accent.
Most Californians are transplants. At least thats how it is here in SoCal. I myself am a transplant from the Midwest and I don't notice that anyone here has accents. Except the obvious ones that are from out of country. People say Californians sound like valley girls because that's what TV portrays. But when I moved to Cali from Kansas some people though I had a slight accent.
Yes, we do. I was born and raised here, and even I know we do. People who live a long the coastline (I mean actually along the coastline, not just kinda near it) have a thicker California type accent. I knew a guy from Monterrey, he sounded like one of the penguins from the movie "Surfs Up". It was crazy.