Barn, stockyard
barn
Barn, pasture, farm, ranch, field are all names for a cow's environment. Most common response may be barn, but only dairy cows spend much time in barns.
An ox is a barn animal such as cow & pig are barn animals.
A 'cow's home' would commonly be a barn, cow-shed, ranch, or etc. They require a shelter to withstand various weather.
Corral, pen, barn, shed, pasture, farm, ranch, etc.
By searching for a cow in a barn or a farm, or even on grazing land.
sheds
Cow house, shippon, cow shed, can all be used.
A cow breathes anywhere it can get AIR, like in a barn, outdoors, etc.
Out in the barn next to Pa's cow
it has a cow, cheese and barn on it.