In California if I'm at a stop in (1) and want to turn left on a busy street, is it okay to use the center left turn lane in (2) to merge right with traffic (as long as I don't use it more than 200 foot)?
Yes. There's a sentence about this in the DMV handbook:
Center Left Turn Lanes
A center left turn lane is located in the middle of a two-way street and is marked on both sides by two painted lines. The inner line is broken and the outer line is solid. If a street has a center left turn lane, you must use it to prepare for or make a left turn, or to prepare for or make a permitted U-turn (CVC §21460.5 (c)). You may only drive for 200 feet in the center left turn lane. This lane is not a regular traffic lane or a passing lane. To turn left from this lane, signal, look over your shoulder, and drive completely inside the center left turn lane. Do not stop with the back of your vehicle blocking traffic. Make sure the lane is clear in both directions and then turn only when it is safe. Look for vehicles coming toward you in the same lane, preparing to start their left turn.
Vehicles using the center turn lane.
When turning left from a side street or driveway, signal and wait until it is safe. Then you may drive into the center left turn lane. Enter traffic only when it is safe.
You can stop in the center left turn lane as well while waiting to merge into the regular traffic lane. You should not stop in nor drive through a dedicated turn lane that might exist in a center left turn lane.
Technically you are not supposed to, but I doubt anyone would say anything unless you were noticeably being unsafe.
Refer to the driver's ed book for the rules on turning. https://www.dmv.ca.gov/portal/dmv/detail/pubs/hdbk/turns