In my opinion, it's never to late to mend a strained relationship. From what it sounds like, you and your mom are having some clear communication issues that are leading to a lot of misunderstanding, as well as a lot of things being said that you don't mean. I think writing a letter is a great idea. In fact, I think doing it right now when you to aren't fighting is the best time to do it. If you write one right after a big fight, odds are the letter is going to sound just as confrontational as the argument was, and nothing will get solved. However, if you take the time to sit down and write this out while you are feeling calm and rational, it's going to be a much more mature and reasonable letter, and will most likely produce better results.
As to the letter itself, my best advice is just to tell your mom about how you feel about your relationship with her. Explain to her that when she says certain things to you (for example, the names she calls you), you get really down on yourself and it really does hurt your feelings. Make sure not to place blame on her for it. Just explain to her what exactly you feel, because there is a good chance that she is just getting caught up in the heat of the moment and saying those things because she's upset. In addition, explain to her that you do really want to follow her rules, but you would also like a little bit of freedom to go with it. You're right-you are a teenager and you do deserve to have a social life and do things that you want to do. That's part of growing up. And I have a feeling that if you explain to her that doing all of that will not interfere with your religion and that you will continue to follow her rules, she will be much more willing to let you have some freedom. Finally, let her know that this is important to you. Explain that all the fighting between you two is really hurting you, and what you really want is a good relationship with her. I would like to believe that almost any mother can sympathize with that, and hopefully she will understand that all the fighting has gotten out of control.
From what I see here, I really think the biggest thing you two (yes, both of you) need to work on is communication. It seems like tensions have gotten so high that she is having trouble dealing with the situation, and insults are emerging as a result of that. This is a situation that is entirely fixable. What you need to do is stop and sort things out as maturely as possibly. One of you needs to take the high road here and mend things, and I think that you writing a letter is a great way to start.
If you want any more help with the letter, feel free to PM me. I would be happy to help. Additionally, if you need someone to talk to about all of this, don't hesitate to contact me.
Take care.
-Isobel