Corral, pen, barn, shed, pasture, farm, ranch, etc.
A 'cow's home' would commonly be a barn, cow-shed, ranch, or etc. They require a shelter to withstand various weather.
A female cow is called a "cow" - if it has not given birth it can be called a "heifer". Perhaps you were thinking of a "cow" called 'Daisey'.
its a called a heifer
crib, stable.
A possible name for a female shark could be "Finley."
Another name for the "father cow" (more properly called a bull) that is used for breeding is called the sire.
The mother cow would be called the dam.
Barn, stockyard
In cattle, a father is a called a sire. Also, it would not be a father cow, because "cow" means female. A male is a bull or a steer.
A calf is a baby cow and a holstein cow is a cow that has not had a baby yet and can not give milk.
The rope is called a Lariat.
It is called the Omasum.