No. A pregnant heifer will rarely come into heat, and her heat period will only be once or twice during her pregnancy, then stop until after she has her calf. But she will not go into heat and start to lactate at the same time she's bred. She can only either be pregnant and have a small chance of coming into heat, or be pregnant and start to lactate (which only happens when she is nearing the end of her pregnancy).
No.
No. Heifers, being young female cattle, don't lactate until after they've had their first calf, which is typically at 24 months of age. A heifer remains a heifer, from birth, until she's had her first or second calf, whereby she is considered a cow. But until then, heifers don't lactate like cows do, and when they do lactate after birthing a calf, they typically produce less milk than their older counterparts.
False heats are rare in cows or heifers that are already bred. If a heifer does go into heat when pregnant, it only occurs once or twice and that's it. But normally, no, a heifer won't still mate when she's pregnant.
A heifer is a female bovine that has not yet had its first calf. A long bred heifer is a heifer well along in the gestation period and due to calve shortly.
A heifer. If she's pregnant she's called a bred heifer.
An "open" heifer is one that is of breeding age, and able to be bred. A "fresh" heifer is one that has recently given birth. A pregnant heifer is, well, pregnant (that's easy). A springing heifer is one who is nearing the end of her pregnancy, within a few weeks of giving birth. "Dry" is not milking and pregnant. "Close" would indicate 240 days pregnant or more. Note that the term has regional variations on the subtleties of it's meaning; so it's best to use it in the general sense only. For example, in many places "open" just means "not pregnant" and doesn't imply that she's ready to be bred.
Yes. "Barren" is just another word for open, or not pregnant.
A cow that has NEVER had a calf in her lifetime is called a Heifer. A cow that has not had a calf YET is a heavily pregnant or heavy-bred, or a short-bred or long-bred cow. A cow that has not had a calf during a calving season is called a barren cow, an open cow, a cystic cow, a cull cow, a meat/slaughter cow, a poor cow, a free-loader, etc.
You could call it a heifer, or a twin heifer if the sibling is also a heifer, or a freemartin if the heifer's sib is a bull calf.
Cow, first-calf heifer, bred heifer, heifer, heifer calf or spayed heifer. See the related question below.
An unpregnant heifer.
A bull calf is male with male reproductive organs, which means he cannot get pregnant. Only females (heifers, possibly heifer calves and cows) can get pregnant. Thus, your question has no merit whatsoever.