All parents, like it or not attempt to indoctrinate their children, in as much as we want to convince them to follow the same belief system or moral code as we do ourselves.
Along the way we have to accept that they will encounter different viewpoints, perspectives and nuances.
Here's an example. I teach my young children that stealing is always wrong. But recently, I bought them a bunch of Roald Dahl CDs (they are aged 6,5 and 3 so probably still just a little bit too little to be able to read them for themselves) for car journeys and bedtimes etc, which they love. Anyway - a line in Fantastic Mr Fox made me think, because when badger questions whether or not they ought to be stealing from Boggis, Bunce and Bean, he reinforces this morality - stealing is wrong. Fox then points out that there isn't a parent in the world who wouldn't steal if it would save their starving children from death.
Would we start out by giving our children this complicated narrative that sometimes, in extraordinary circumstances, stealing, while not a good thing to do, can occasionally be acceptable? No. We give them the absolute and expect them to accept it and it's only later on, can various ethics be explored, once they have the intellectual capacity to do so.
So (sorry philosophical ramble) I'd argue that all little children are natural theologians, in that they are interested in the questions of who made me, who created the world, what happens when you die and unquestioningly accept the idea of God, when it's presented to them.
It sounds like YABU and intolerant because you can't accept that your daughter may think differently from you and you want to shed a whole heap of convenient blame on your MIL.
The anger seems to be a combination of things. The history of the bad relationship with the MIL and the fact that you feel that you might have lost control of your daughter's impressionable mind.
I am a committed Christian, I disagree with a lot that your MIL seems to be putting forward. What concerns me most is the description of people as cockroaches, which is an extremely un-Christian attitude.
The calling you a whore, while deeply unpleasant, I wouldn't take too seriously, in that though it's a sexually charged misogynist term, sounds like she totally lost her temper and meant it more in the sense of a dirty slattern etc. Someone who she felt was all over the place. I am not excusing it, but just think it doesn't sound like it was meant in a sexual way.
It is unreasonable to expect her not to want to see her grandchild and also to want to limit her religious and individual freedom, because of an ungrounded phobia of indoctrination.
Reading about the history between you, there's always two sides to every story and my honest assessment is that you are both a pair of self-righteous neurotic control freaks with anger-management issues. Have you actually told your MIL that you'd prefer her not to use said mug? Is it really worth getting upset about a used teabag?
It's your DH and your daughter I feel sorry for, caught between a nutty, but probably well-meaning reverend who reads too much Daily Mail, and a neurotic mother who is evidently extremely pleased with herself for taking a week off, not to spend with her young family, but to do some virtue signalling and pity tourism, (sorry but your posts reek smugness) expecting her partner to pick up the slack, while at the same time not wanting him to take any annual leave either.
If I went off volunteering for someone more worthy of my time than my impressionable and vulnerable three year old, knowing that my partner wasn't going to be able to care for her, then I'd be blooming grateful for whoever stepped in to provide free childcare and not be so entitled as to rage that they dared to share their belief system (which presumably brings them great comfort) with their precious granddaughter.