Frankly, I can't understand the obsession with some grandparents (both maternal, and paternal), who seem desperate to have their young GC stay overnight! I'd be more than willing to do so, if my babysitting services were required for some occasion, but it's not something I would request to do, just for the sake of it! What is the point, exactly? I can't think of anything worse than having a fretful child, waking in the night, in different surroundings, wanting mummy, or daddy.
Obviously, it sounds like your MiL has become a bit of a nuisance, turning up unexpectedly several times a week, and constantly suggesting she be allowed to have the child on her own. It would drive me insane. It seems she has plenty of time to see her GC, but you can't allow her to dictate to you what she wants, or how much time she gets to spend with your child.
I disagree with what some posters have suggested ... I don't think you should have to lock yourself in your house, and pretend not to be in, just to avoid her. And, likewise, you shouldn't have to go out when you don't want to!
Other than saying your DH is a bit soft, you've not said what he actually thinks of the whole situation. Have you discussed it with him in detail, telling him how annoying/pushy his mother is becoming? Is he aware of the full extent of the problem, or does he think that there isn't a problem ... that it's his mother, and he's OK with her spending so much time with his daughter? If he's working, and not around when she visits so frequently, maybe he doesn't understand how annoying it is. The only way to resolve this problem is to make him understand what's going on, and his mother needs to be told that you don't need any overnight babysitting right now, but in the future, if/when you do require any babysitting, you will let her know. Just tell her that, for now, it's not something you need, but maybe when your child is older, and you need a babysitter, you'll let her know.
You do sound very young, and perhaps lacking confidence to assert yourself, but if you don't say something soon, matters will only continue to get worse.
As for the frequent visits, maybe you could tell her you're going out shortly, and if she says she can look after your daughter, just tell her there's no need as you're visiting a friend who also has a little one, as your daughter needs to start to socialise with other children. If she doesn't get the hint, start to pack your bag, and get your coat ready.