Your fundamental premise is flawed. Bats are not descended from rats. Bats and Rats had a recent common ancestor, but they both appeared fairly close to each other (well, rodents and chiropterans did anyway, not the modern species of bat and rat). Monotremes (of which there are only 3 species, the platypus and the long and short beaked echidna) most likely represent a type of animal closer to the original mammalian ancestors - the placentals and marsupials would have split off from monotremes first, then themselves further split down the modern lines. All of this seems to have happened fairly early though, as we have fossil evidence for Monotremes, Marsupials and placentals here in Australia (not counting bats there, for obvious reasons).
Stealth I'm going to ignore because he should know the answer by now, and it's impossible to change the willingly ignorant.
Felines and Canines are relatively close (both in order Carnivora), but so are weasels and ferrets (they're not rodents - think about their teeth and it's pretty obvious). Actually, the general distinction in carnivora is a dog-like/cat-like split. I'm not sure how closely that's linked with fossil or genetic evidence, but it would suggest dogs and cats are actually descended from the two branches that diverged very early on in carnivoran evolution. NB - weasels and the like are actually in the dog-like group, so they're not the closest to cats by any stretch.
High Max - Note that "Shrew-like" and "Rodent-like" should be taken as very different things. Shrew-like is morphological, while rodent-like implies some kind of genetic relationship. Rodents are a relatively modern innovation, post KT mammalian diversification, and are quite highly evolved (again, think about the teeth - those big, constantly growing, constantly sharpened front teeth to be precise).
