I have some extra jump/kick boots from polypiling that I'n considering equipping my pet MMF with. I'd have to enchant them a bit to induce a trade for its +2 Iron shoes (probably FooProof them, too). That's a bit much if they won't get used in the manner they're designed for. So, are pets able to make use of any of these boots?
Based on a bit of testing and source diving, I'm fairly sure that pets (or monsters in general) don't benefit from waterwalking or levitation boots. Giving those to your pet (master) mind flayer would be pointless anyway, since mind flayers can fly.
I also suspect that monsters won't use jumping or kicking boots. Indeed, AFAICT, there does not seem to be any code for jumping monsters at all, and while monsters can indeed have "kick" attacks, those are just yet another natural attack type, and don't seem to be affected by kicking boots in any way.
Monsters do, however, benefit from speed boots, but not as much as the player. Indeed, for monsters, there is no difference between intrinsic speed (gained e.g. by zapping them with a wand of speed monster) and extrinsic speed (gained via speed boots, the potion of speed or the spell "haste self"), so you're better off just zapping your MMF with the wand (which is pretty common, so you probably have some spares unless you polypiled them all away).