First of all, kudos to you for leaving a violent husband and getting your children out of that situation.
Secondly, I get how frustrating it can be to have a parent criticize your childrearing. Sometimes parents figure that their kids are grown and on their now, so their job is done. It's hard for them to have their kids come back, especially with their own kids, and have to go back to childcare themselves.
Third, it's great that you are going to school. Medical assisting is generally considered a promising career.
I'm a little confused - you say there's a problem if you "leave the kids with her or at home with someone else" - does this mean that you live with your parents or that you live elsewhere and either leave the kids at your place or with her at her place? The other responses seem to have the feeling that you live with your parents and I thought that too, but this one line is a little confusing. You also say you are job hunting, but say your father asks you "How's work" - so do you have a job or not?
If you do live with them, I'll say first that your mother is being a little dramatic about not going to the bathroom. A 6 and a 3.5 year old can stay alone for a few minutes. It's also frustrating when you have different standards of childrearing.
However, beyond that, she has some valid points. You aren't interviewing 8 hours a day, and while you have work to do for your degree, you do that when all other mothers do their work - when the kids are in school or in bed. During the day, when you are not interviewing and if you are not working, you have to do a lot more with and for your kids. That means that they go on errands with you. Is it a pain? Yes, sometimes. Maybe often. I can't imagine that you have very many doctor's appointments and in general, unfortunately, kids go with you. The exception might be a GYN exam or anything where you need X-rays (like a mammogram) but you are too young to need much of that! So the kids go. You learn to take books and small games to keep them occupied. I can't imagine why you would have to do that many things child-less. If you organize yourself tremendously (which is a skill you will need as a medical assistant, a job that requires efficiency and accuracy), you can combine errands into one trip.
Housework - if you're in your mother's home, you do it to her specifications. Maybe your own room is up to you, but public areas are up to her standards. 2 kids make a lot of mess and use a lot of dishes and laundry - so those are your responsibilities. If you aren't working and you aren't in school during the day, I'm not clear why you don' have a lot more time to get things done. I imagine your oldest is in school, at least a half a day, so you can take the 3.5 year old out of your mother's hair. You can also teach both children the basics of tidying up their own mess. They can take their dirty dishes to the sink, pick up their toys (maybe put a bin or basket for each child in the family room to control the clutter, but they do the work), sort and match clean socks, help make beds, etc. Time to teach them to be independent - you don't want them getting stuck in a marriage that's unhealthy, do you? Now's the time to start!
If you've only been apart from your ex since March, I don't understand why you have a boyfriend that the kids "love"! I understand you may want to date and be treated nicely after being treated so badly, but you don't introduce a new man to children who have undergone such a big change and who have moved in with grandparents! And you don't push them or even think about them "loving" a new man! If the excitement of a new relationship is taking you away from your children and your responsibilities in your parents' house, then you are making a huge mistake. Your kids come first, and so do the people who are putting you up. You can, if your mother's house is picked up and the kids' laundry done and so on, have 1 date night a week, but otherwise your focus is on the kids and your education.
If you are working and studying and raising 2 kids, you have to learn to prioritize, to be way more organized and efficient, and limit outside distractions. Make a list with your mother of what she wants done in priority order, and set aside a schedule (working with her and your father, who seems agreeable) of when these things will be done. Maybe if your mother sees that you have a time to do them, she won't nag you so much about being late. I know you want your own time, and errands can give us moms some breathing space. But you aren't going to get much of that right now until you finish school and get a job (or better job). When your mother says you are living your life while she watches your kids, do you hear an element of truth in that? Are you out doing fun things (boyfriend, friends, activities) instead of being in the house, overworked, like the rest of us?
I think some re-ordering of priorities will help everyone.