I'm talking about animals we can pet and cuddle with.
Dictyostelium discoideum (AKA slime mold). Cute and cuddly is in the eye of the beholder. Dictyostelium discoideum can aggregate and move around like a tank tread.
Birds, Bats, flying insects don't have propellers, but when is the last time you saw an airplane alight on a tree twig or electric wire?
Saurischian dinosaurs have an axle-in-casing hip joint structure - the legs can swing front-to-back but not side-to-side or twist like our ball-in-socket hip joints. You couldn't outrun a T-Rex but you could outmaneuver him/her/it.
a wheel-like device can't be innervated or be supplied with blood without wrapping the pipes around the axle.
Not true. The crank shaft in your car is a fancy wheel and has passages for supplying oil through the connecting rod to the piston. Electric motors have brushes or bushings for passing electric current to the rotor (a wheel) and these could also be used for passing signals as they are in rotating radar antennas. Not sure how these would work for nerve signals which are a combination of electrical and chemical.