I'm sorry, but I think your DH has his head stuck in the sand. It is a problem, but maybe because your mother has never talked to him directly about it (I'm guessing), he just ignores it or doesn't realize what it is.
First, set up boundaries. MIL cannot just "stop by" without a call. Make it known, in front of DH and her (if DH won't take the bull by the horns, so to speak). Tell her that there is too much going on/you're too busy/kids schedule too hectic/you value your privacy though you love her very much (whatever you think will work best). Then tell her that she needs to call before stopping by, because it puts your family in an awkward/uncomfortable position if she doesn't. And when she does stop by without calling first, you can do one of two things: 1) leave the door locked and don't answer, or 2) (which is less confrontational), grab the kids and head out to the store, and say, sorry, we're on our way out the door, next time please call. And keep doing that until she respects the rule.
Second thing--find out when your mom is coming, and tell your MIL, point blank, that since your mother lives so far away and doesn't get to see the kids very often, she needs a little bit of one-on-one time with the grandkids. Tell her that since she sees the grandkids every week, you know she understands how special the bond is between a grandma and the grandkids (say it with a smile, even if you're being inwardly sarcastic!). Make sure your DH is aware of this and is "okay" with it. Then, if MIL shows up during the special time with the kids,
Thirdly, you might want to directly address this with MIL: "I've noticed you have a hard time sharing the baby when X is over visiting. Then, wait and let her respond. Later, bring up the gifts: "And it seems that every time you visit, you bring 2 gifts for each kid. We want the kids to love you for you, not your gifts, and we're concerned, that being, young, the children will only learn to be selfish, or expect a gift each time they see you, when we'd rather that they look forward to visiting with you and not focus on the gifts."
She may not be happy you're bringing this up, but you'll be doing so in a kind manner (say it with a smile), and you'll be indirectly telling her that you're aware of her competition.