I can't help with the doggie soiling issues. (although our carpet shampooer did a great job on our rugs when you dog was young.) You might want to call the Stanley Steamer people and see what they recommend . . . they do wonders with getting smoke smells and stuff out of homes . . . ???
Your mother-in-law is another issue. Bottom line is that you will never change her, so you have to accept her the way she is.
Money? Don't carry any around. That way you won't have to foot the bill. Be prepared. When she suggests you pay and she will pay you back, you can say, "I would, but right now we don't have the money either." And stick to it. It's not worth funding stuff and then feeling used, hurt and angry because she didn't keep her word. You already know she won't, so change yours from "Okay," to "nope. I can't do it either."
The other thing is to change the expectations. She has been who she is for a long time. She's not apt to change now. The good news is that you don't live with her, you live with her son. :-) So you have to visit periodically. You can survive that for a while, then get back home to where you can really enjoy family life.
Knowing what you know about your mother-in-law you can help your kids as they navigate through their relationships with Grandma. Encourage them to enjoy being with her, for her good points, and help them when they get hurt, to see that all people have good and bad points, and we enjoy the good ones and have to learn to accept and forgive them for the bad ones.
My guess is that the real reason you didn't get the coveted toy back sooner is because it had already been damaged and she didn't know how to admit it and return it to you the way it was. It was much easier and certainly less confrontational to simply forget about it, and hope you did, too. But you asked enough that she finally had no choice, so she mailed the problem to you. It is heartbreaking, to have to deal with it, and it's difficult at best to have to deal with your own hurt, anger, resentment, etc., as well as to see the joy your daughter had in this toy fall apart.
And only you can decide whether or not it's worth all your efforts to fix it, or to simply give up and let your daughter think it got lost at Grandma's house. (Has she seen it?) Some hurts of childhood we can't change. We have to cope with them, and cope with our own emotions over them, but we can't change them. This is a very painful lesson, for both of you, but now you know that you should never leave anything of value at their house. That doesn't make them bad people; it makes them careless people, or people with different values from yours, but obviously they at least raised their son well, so there's still something good there. :-)