No, don't think you're bonkers LS, but you do obviously have different ideas and values to your MIL. That's not necessarily a bad thing, it just means you will judge events and so on differently.
I think her reaction to the cat should be considered separately to her relationship with your DCs, because as I think someone has already said, they are not really connected. I'm sure she is genuinely very upset about the cat. When one of our cats dies last year it was awful and I was certainly not in the mood for seeing anyone the day after. The card's a nice idea. You could also offer to help her take the cat to the vets for cremation or whatever she's going to do (if that's practical for you) if you want to help out. I also think that the cat may be a focus for her guilt/grief over her own mother's death.
Btw, I think it's not entirely fair to categorise this as 'preferring animals to people' - I prefer people to animals, but at the same time I'm very attached to my pets. Don't forget it's not a choice of saving the life of a child or a pet (which I think was your comparison) it's between grieving the loss of a much-loved pet and seeing children who will be there another day. I think I'd make the same choice tbh.
Sounds like it was already established, well before the cat incident, that she doesn't want the same relationship with your DCs that you would like her to have. Or perhaps her desire for that is outweighed, at any rate, by the phobias and anxieties and other problems that are presented by having that kind of relationship - she sounds as though she has a lot of these, what with her husband's problems and her own oddities. So really, your best option is probably to accept that, since you can't make her relate to you and your family in the way you want. The only other option would be to do it on her terms, since (to be blunt) you are more bothered about keeping up the family stuff than she is.
It sounds as if your DH is quite frustrated by the way his mother operates. Has he talked to her openly about it, or is it all done indirectly (or does it fall to you to try and work out a solution)? There is also the option of saying to her directly:
'We'd really like it if you'd see more of the DCs, but it doesn't seem to be your thing. Tell us honestly: is it that you just want a quiet life [giving her a get-out here] or would you like to see them more but there's a lot of stuff that gets in the way? Because if that's the case, we can try to work round it'.
The thing with this though, is that you will have to a) accept it if she says she doesn't want to, and b) take more of the strain yourselves in accommodating her if she says she does want to see them. So there's no great solution for you, but it might clarify where you stand.