Is it legal for them to mail me the ticket instead of giving it to me at the time they arrive at the accident?

The police can make a determination based upon what information they gather at the accident scene. There may be some comparative negligence on the part of the other driver. The insurance company determinations are only for the purpose of what they will pay and are binding upon no one but themselves. You are at fault for making a left turn when it was unsafe to do so even though the other vehicle was speeding you could not enter the street until you saw that it was clear to do so. You will not have to pay for the cost of the emergency vehicles coming to the accident scene. There normally is a set amount for each type of traffic violation, you should be able to look it up in the penal or vehicle code or contact the local traffic court clerk to find out ,what that amount is. At the initial hearing you can request a trial to try to get the fine reduced, but it would not be done so on the basis that the other driver was somewhat at fault. The ticket does not have to be given to you at the accident scene, especially because the officer probably had to go to the hospital to speak to the other driver and passengers so could not give yo the ticket then.

You are required by law not to start or stop or turn left without seeing that you can do it safely so when you cant turn safely you usually get a ticket. Insurance co will sort it out

Yes it is legal.

Yes, and it is common.

It is common to get the ticket in the mail. No, you do not have to pay for the police and fire-fighters. The insurance company may battle who was at fault, but in your case you are. If you check the statues, the person making a left-hand turn must yield to oncoming traffic. Thus, why you got the ticket.