The near hands thing makes sense. There are anthropomorphized dogs and cats who have their paws modified into hands, but it does require a more cartoonish treatment of them.
Mice are about as small and helpless as a mammal can get. They're nearly at the bottom of the food chain, so to speak, and we tend to like stories about the "little guy" making good, and such stories are probably even more relatable to kids. And while they're pests, house mice also live in a hidden world that's next to ours but mostly out of sight, so that captures our imagination in the same way wainscot fantasy worlds do. Oh, and there are so many different species of adorable mice (or mousy creatures) that don't live in our pantries too): voles, deermice etc.
I can remember popular anthropomorphic mouse stories from my own childhood too: The Cricket in Times Square (the focal character was actually a mouse who lived in a subway station in New York and befriended a lost cricket from the countryside), Robert Lawson's Rabbit Hill books (the rabbits were the main characters, but there was an important mouse too), Mrs. Frisby and the Rats of NIMH, Robert Lawson's Ben and Me, told from the pov of a mouse who lived in Ben Franklin's hat.
And of course, there are no shortage of cartoon mice, from Disney's Mickey to Jerry (of Tom and Jerry fame), and plenty of others.