I have no idea if you are still reading. I hope you are.
I have a few thoughts - but first off I would like to say I respect your decision not to go to therapy. I get that not everyone can afford therapy or believes in therapy - it simply means you will have to find other resources.
There are (in all probability) free or close to free groups for parents of teens. You could check those out. It might be nice to have someone IRL to vent to.
As for your post - here is my take - YMMV.
There are numerous types of posts on mothering - some looking for support and some looking for advice. Yours (and I have read the OP twice) seems to be asking for advice. I don't think you should get angry or frustrated over lack of support - when it was more of an advice post, kwim?
You seem to be angry - period. Angry at your mother and DD, and they are angry at you......anger, anger, anger. You really cannot fix them - but you can work on yourself! There is a book that has been recommended called the dance of anger - I have not read it, but heard good things. Perhaps you could check it out?
As per the specifics of your post:
I would not ask her to babysit - at all! I know it sucks that you cannot ask her to babysit - but you just shouldn't. There seems to be a lot of score keeping going on - you have demands on DD (babysitting, grades) and your mother (ummm - watching your child, apologies, holidays) and they make demands on you. Enough. Someone has to stop the scorekeeping, and it might just be you.
I am not sure what to think about her living with grandma. Part of me is completely against it - you do not like your mother for goodness sake and are letting a child live there, yet another part of me acknowledges that she is 15, and it might be better that she has some say in where to live. I just do not know - it is not an ideal situation.
Last thought - you say her grades are horrible. Is school working for her? Can she Homeschool or switch schools? It just seems that she must be miserable - not getting along with family memebers, not doing well at school - if there was one thing you could fix it might help with her quality of life.
Peace and good luck,
Kathy