Am I at fault if a pedestrian ran across the red light in the cross walk in the path of my car?

If the pedestrian truly darted out in front of you against the red light, such that you could not stop in time and avoid hitting the pedestrian, much if not all of the blame should be put on the pedestrian. There might be a tiny bit of comparative negligence placed on you for not stopping and failure to keep an eye out for pedestrians. A lot depends on accident reconstruction.

California law does not use an "all or nothing" approach to fault, or what lawyers call liability. In California fault is on a sliding scale, often referred to as comparative negligence. The answer to your question on how to divide fault between the parties will depend on more facts that what you have provided in your post. I suggest you turn this over to your automobile liability insurance company.

Maybe. You have to keep your car under control at all times and see what there is to be seen. Sounds like the lady who ran in front might be negligent too and if so she could not collect. If the light is red for you and she is in the crosswalk isn't that where she is supposed to be? What have I missed here?

To some degree, probably. We have "comparative negligence" in Florida, so no one person has to be 100% at fault. If she was in the cross walk crossing against the light, she would probably be found to be at fault by the vast majority, but I can certainly see a jury putting some small percentage of fault on you, as well.

No, as long as you did everything you could to avoid the accident.

Well, it sort of depends. If you are in a residential neighborhood, and a child runs out in front of your car, you may be at fault if you could have seen the child before he/she reached the street. With an adult the rules are a bit more in your favor, because adults can be held responsible for their own actions. Along a public road (not in a residential neighborhood), it would seem to depend on whether or not you could see the pedestrian and could reasonably determine if he/she was going to enter the roadway. If a person is standing there, and then as you get to the spot, the pedestrian suddenly darts out, then you probably would not be at fault. What makes is confusing is that you need a license to drive, which means that you are obligated to obey the rules of the road. A pedestrian does not need a license to walk. The only law I can think that would apply would be the one that prevents someone from "interfering with traffic."