Typically, insurance follows the car first and driver second. This means you, the owner, will have to use your insurance as primary and the driver's insurance (if they had it) as secondary or excess. Some insurance companies might have exclusions about covering some people driving your vehicle so contact your insurance agent to make sure your friend was covered and if so how to go about filing a claim.
So the answer is yes a friend driving an insured car is typically covered if given permission to drive, unless exclusions on the policy restrict the coverage. If the person driving your car doesn't have insurance on their car, then no they would not be fined while driving your insured vehicle.
In California the penalty for driving without insurance can be a fine or license suspension or both. A first offense fine will cost between $100 and $200 while a second offense will cost between $200 and $500. Have our friend call the court or entity listed on their citation to find out what they need to do to handle the situation. The court is likely to decide if the fine is sufficient or if their license will be suspended as well. If your friend or you are in need of California Auto Insurance we can help get a policy so they can show proof of financial responsibility (car insurance).