You and your family don't like your mother in law do you?
The first birthday, I was with you, and the huh like he'd know or what the hell does it have to do with what other people think, but I can also understand Anna wanting to see their grandchild on their birthday.
Second one, it starts to be all about your family, they seem to trump her, I'm sorry but she's your sons family. Maybe she's of the opinion older people should have a seat while your of the option first come first serve, in which case prehaps you as host should have said ah my sister is sitting there mother in law let me get you another chair.
Maybe she thought because the relative that wasn't there, wasn't there they'd not been invited, I think it's fairly normal to call a family gathering to celebrate a birthday a party, again maybe you could have helped that situation by going "shame so and so couldn't come but their out of the area"
Third one yes I like wellandtruely I'm confused to who conplained about who, but again it cones over like your sister and mother are saints while you can slag off your mother in law. You expected your mother to come in but didn't like your mother in law there. Their both your sons nan.
I might agree with ignoring your son especially on his birthday is not on, especially if she's going to make a fuss about seeing him on his birthday.
Maybe because you'd made it clear she wasn't welcome on his birthday she didn't feel like she'd be welcome at the meal, were your family going to be present at that? Maybe she felt that she'd be better of playing bingo than being made to feel uncomfortable with your family.
I say this as some one who's nan on my fathers side and my half siblings nan on their fathers side, actually could be the mother in laws from hell, so much so that my half siblings refused to see their nan from the age of 6, but the tone of your post seems to display how much you don't like your mother in law and it's all about your family.
Sorry