Yes, they do. For most breeds, it’s DNA test on hair samples, done mostly at University of California Davis, a preeminent center for horse genetics studies.
Yes, it’s called a ‘teaser’ horse. But it’s a stallion, not a gelding (unless you have a proud cut one) – usually a young stud horse. The teaser stallion is used to test the reaction of the mare, to determine when she is in full heat and most likely to conceive, and least likely to fight the stallion off.
You generally just lead the teaser stallion up to the stall containing the mare, and let them sniff at each other through the wall, and talk to each other. If the mare’s reaction indicates she is in full heat, then the teaser stallion is put back into his stall (frustrated!) and the mare & covering stallion are prepared for breeding. The teaser horse is not actually in with the mare, and certainly never mounts her – there is no need for that, her reaction tells you all you need to know.
And a properly gelded stallion loses ‘the urge’ a few months later, as the testosterone hormone level in his body drops. That’s the point of gelding the horse – to remove the stallion behavior urges, and make him more usable. (Occasionally, an older stallion that had extensive breeding experience will still exhibit stallion-like behavior after gelding; but that is basically just showing learned behavior. This is what he was used for all his life, and what he thinks he is supposed to do. It eventually dies out, due to the hormonal change of gelding, and just the increased age of the horse.)
Yes, twins are much rarer in horses than humans.
Presumably, the twinning trait has been largely bred out in horses over years in the wild – it is much more dangerous for the mare, and much harder for her to successfully raise 2 suckling foals at once. But it does seem to run in certain bloodlines (just like in humans).
Twins are still quite dangerous for a mare, even with good veterinary care. In fact, if twins are noticed by ultrasound, it’s common for some breeders to have their vet identify the less developed one and ‘pinch off’ that twin, so it never develops into a foal. This can be done only if it is noticed quite early in the pregnancy.
In sixty-some years of breeding horses at our farm, we have had only a single set of twins. And despite some really strenuous efforts, we were only able to save one of them. And that mare never developed fully; she was a very short, small mare all her life. (But extremely friendly to humans, probably because of the extensive care she got starting from birth.)
I don’t know how far back that was, but for the last 15 years Quarter Horse stallions have to have DNA info on file with the registry. So if questionable stuff like this went on, it could be caught fairly easily if they looked for it.
Of course, with the number of Quarter Horses registered each year, nobody bothers to look for it in most cases. But if the horse became prominent because of winnings, it might be looked into by the registry. Especially because it’s really hard to do such things in secret, and somebody ends up talking, and rumors like this spread real fast in the horse world.