It is certainly reasonable to ask your MIL to keep the place reasonably tidy, to stay out of the master bedroom, and to throw away diapers. However, I'm not sure how likely they are to get what they want.
Is the MIL's home normally untidy and/or dirty? If so, this is likely just what she's like, and the chances of changing are small, and the couple are going to have to find other arrangements if they want it done differently.
If the MIL is normally relatively tidy, I can see two likely explanations. One is that the MIL isn't happy with the situation, and the mess comes from that. She may have thought she'd love babysitting, but realized that she's not happy with doing it every day, but feels she can't step down because of how it's needed. The other possibility is that she's enjoying it, but is not able to watch an active one year old and keep up with the mess, and is perhaps letting him do whatever he likes, because dealing with a toddler who is screaming because he wants something is exhausting.
If she's retired, she's likely over 65, so it may well be that she physically can't provide childcare up to the standards of a younger nanny, but would be fine as an occasional babysitter. My mom loves her grandkids dearly, but from the beginning has freely admitted that she wouldn't be able to provide regular babysitting.
If the MIL is fairly reasonable generally, I'd approach it from the perspective of "I know how hard it is to keep up with the kid, so this is how can make it easier." For example - leave a prepared lunch and snacks in the fridge, so she doesn't have to make food, and minimal dishes are used. Put a baby gate on the stairs, and arrange things so she doesn't have to go up there at all (ie, the baby can nap downstairs). Pack away a lot of the extra toys and dishes so there aren't as many to make a mess with (if there is only one sippy cup, it has to be re-used). Leave some approved videos so that the MIL can park the kid in front of the TV to get breathing space when needed.
Otherwise - I agree with PPs that half an hour of tidying a day is a really good deal for a free, full-time babysitter, although the dirty diaper and wipe thing needs to be addressed. Ultimately, though, it does come down to beggars can't be choosers - they can take what they are offered, or make their own arrangements.
The one thing the mother can do, though, is make sure the father shares the work. Clean up the immediate stuff, like dirty dishes and diapers, and leave the rest for him.