Sometimes our own parents have a hard time stepping out of the role of being the center of our attention, and may desire to have lots of control in order to feel secure with the new state of things, esp. now that we have children. Although we may have changed monumentally as adults through our transition from 'person' to 'parent', our own parents may be less emotionally equipped to deal with this change. Their connection with their pregnancy and early parenthood has faded, and the propensity to try to relive the "family" dynamic you grew up with is very strong. (This is why so many of us dread going home for the holidays...we have that feeling of not being allowed to be our own adult persons, instead our parents revert to treating us like children.)
Sometimes our parents are able to flex and change with us. I can see how you feel that your mother doesn't respect your schoolwork or your preference to keep the boys at home, where parenting and visiting with others is much less work than being at someone else's house. And your anger at feeling disrespected is natural and healthy.
On the other side, her desire to control the situation by being at her house is also natural. I know lots of grandparents who insist that the kids playing at their house is fine when it isn't. I think it stems from a feeling of insecurity; if she feels that she can be on her own turf and call the shots (being in control of your coming and going/transportation), this can give her a sense of "normalcy" in the parent/child dynamic.
Depending on where she is, it might be worthwhile to arrange a visit with just the two of you in a neutral space to talk about the relationship troubles you two are having. Perhaps explaining to her that you would like her to be proud of you for pursuing your studies and caring so well for your boys, instead of feeling that these things are coming between the two of you. Also explaining to her that this situation with the car isn't 'forever', nor will the boys be so very little and needing so much in regard to attention and familiar playspaces 'forever'. Putting it in the light of "this is me and my family growing, and I'd like your support" might help. And then just talk; acknowledge her feelings and ask if you can't come to some compromises that respect what you and your family are needing at the moment. Maybe that's a half-hour phone call one night a week where your partner watches the kids and she can have your attention...the two of you will have to figure it out.
It's essential to your relationship that you tell her how you feel before there's more tension or a big blow up...those are hard to come back from and feel good about. So asking her for some insights/alternatives to what's "not working" right now and help to find something that does work for both of you will help. Yes, boundaries are important, but then give her some room to come back with some ideas that might work for her. Growth between parents and children is hard at any age, but invite her to do this with you.
And by all means, let her know that while you love her very much, you don't like what's going on in your conversations. And give her some very concrete examples (as in-"When you said XYZ, it really hurt my feelings. I'm doing the best that I can and it feels like you don't recognize this; only that I'm letting you down in some way.")
So sorry; parent stuff is so heart-hard to deal with sometimes.