Yeah, it's tough when you can pin anyone down about their expectations and plans.
Just trying to picture what happens here--you need to go grocery shopping, DH is at work, you have to take your toddler because MIL doesn't babysit. So the question is, do you leave MIL at home alone, or offer to let her come with you? One question I have is, what does she do when she's home alone? Is she the type to enjoy the downtime and get some reading done, or would she boredly watch TV for a while then start snooping in your closets?
Obviously if she can't be trusted
at home alone, that's bad, and you would pretty much have
to take her with you every time just so you don't worry about what she's getting up to. But if it's just a matter of her being bored, I don't think it's rude to let her stay home sometimes. I think it's reasonable to expect that adults can entertain themselves alone on occasion, especially if she's staying with you a long time. (If she was only staying three days, I think it would be weird to leave her home alone, but multiple weeks is different.)
So I think you could say, "Toddler and I are going to the store, we'll be back in about two hours. Help yourself to the XYZ!" and then bustle out the door. Not every
time, but maybe every other
time. Maybe you could even say something like, "MIL, I feel bad dragging you along to these boring errands with me and Toddler. Have I shown you how the magic Netflix box works? Could I run you to the library to check out some books?" In other words giving her suggestions about stuff she can do at home alone, while you're getting the normal chores of living done.
Also, is she capable of getting out on her own to do things by herself? Are there places she will want to go on her own, or special places you can all go as a family? If you think she'll do it, you could come up with some suggestions of places nearby she could go on her own, and encourage her to do so--like, "The museum exhibit is open at these times, you can take the Blue bus to get there, I'll drop you at the bus station on my way to the grocery store..." Or, if she won't go alone, you could plan some all-family outings (you, MIL, Toddler, and DH), and get your "togetherness" then, so that when you leave her home alone while you grocery shop, no one will be able to say you didn't spend any time with her.
Regarding your DH's general expectations about spending time with her--that's a tricky one. Maybe you could proactively plan (and tell him about) blocks of time when you will make yourself scarce, "so that he can spend time with his mom." As long as they're in his general "free time," shouldn't be a problem, right? And then you get some time away from her, even if you have to leave the house to get it.