To replicate this, you can't expect to get away with only replicating the complexity of the mouse. You potentially need a computer with as much complexity as the evolutionary algorithm which led to the mouse's movement algorithm.
Only for some cases. I've watched a horse being born and seen pictures of other newborns. The foal can stand within minutes, and the right sequence of moves is clearly built in because it works the first time. Newborn horses can walk within hours and run with the herd within days. Lying down, though, is not pre-stored. That's a confused collapse for the first few days. There's an evolutionary benefit to being able to get up and escape threats early, but smoothly lying down is less important.
Most complex organisms spend some time inside a protective shell of some form before getting out into the wild, where I suspect they learn to coordinate their orientation sensory organs with their limbs to produce necessary movement patterns.