When a female has her second calf (baby), she will then be called a Cow. The word cattle is speaking of bovine as a whole, including babies, females and males.
You usually slaughter beef cattle around the age of one year, at this time they have the ideal amount of conditioning (beef).
Beef cattle raising is just a play of words for a job of raising beef cattle. Raising beef cattle often involves breeding beef cows to a bull to produce calves that are sold for the meat market. However raising beef cattle also involves raising purebreds to sell to other producers; stocker/backgrounding operation which "raise" weanling calves from weaning age to adequate age and weight to start finishing; and "raising" steers or finishing cattle to slaughter.
beef
Raising cattle for beef (or, as you like to call it, "beef cattle raising") has been around since the New Stone Age, which is over 10,000 years ago. Thus there is no definite year when such an event was "invented."
According to the Merriam-Webster Dictionary (see link below), Stock Cattle are "all cattle other than beef cattle and steers over three years of age, [especially] cattle of breeding age."
On a $/lb basis.
Pretty well any age, really. Bobby calves (those that are a few days old) can be sold for veal (but this is primarily limited to dairy, not beef cattle). Baby beef calves are those that are from 1 to 6 months of age. Beef steers/heifers/bulls can be slaughtered/sold between 6 and 18 months of age, if your purpose is for beef production.
A sheep can be culled from the flock at any age. Generally culling means that the sheep is being removed from the breeding herd because of age, something wrong in the genetics, too closely related to other animals in the herd or illness, not necessarily strictly for being killed for meat. Culls can become someone else's pets! Lamb that is grown specifically for meat are processed before one year of age. Sheep older than one year are processed as mutton.
There are too many variables at stake here: the age of the growing cattle, what type/breed of cattle (i.e., dairy or beef), the type of feed, etc. But basically there are some rules of thumb regarding protein requirement for growing animals: Cattle that are younger than 6 months of age require a feed with 16 to 18% Crude Protein. Growing cattle around 8 months need around 14% CP, and yearlings require around 12% CP. Fattening steers over 18 months of age require between 10 to 12 % CP.
At 2 years of age. This is when all the baby teeth fall out and are replaced by adult teeth.
Brahman cattle can live over 15 years of age, often up to and over 20 years.
Market steers and heifers - those cattle that are bred and raised for the beef industry - are typically sent to market at 20-30 months of age. However, up to 20% of steaks sold in groceries across the country come from cull animals - aged dairy and beef cows as well as bulls. These cull animals may be as old as 12 years when presented for slaughter.