Sit, stand, and lie are verbs that all refer to the prototypic human body. They represent three basic body orientations: respectively, vertical, folded, and horizontal.
These verbs have a number of common peculiarities:
- they are all very, very common
- they are all intransitive
- they all describe a state, but allow the progressive - He's just standing/sitting/lying there.
- they are all irregular - sit, sat, sat; stand, stood, stood; lie, lay, lain
- they all have irregular inchoative/causative related verbs: set; stand; lay
(parenthetically, this is where the sit/set and lie/lay problems come from)
Combined with an attributive predicate
A that can describe the human body, any of these verbs can mean 'sit/stand/lie in a(n)
A manner'. So sit still, stand rigid, lie limp, etc.
As far as other animals are concerned, if you can metaphorize a human body on them, they can sit, stand, or lie. But not necessarily with the same body parts -- when a dog or cat is standing still, it's likely using four legs, not two; but when it's standing up it may well be using only two.
It depends, in other words, on your metaphorical image. And metaphors are always literally incorrect, since they violate the Law of Contradiction.