1. Yes, they are genetic. However, you still have to get her TEG rolled at the trait genetics board to find out if your joined horse is TEGTEG, TEGteg, or tegteg. You can read more about the trait genetics and what they mean here:
http://www.beqanna.com/forum/breedingtraitgenetics.php
2. Her foal would inherit the immortality, either as expressed or carried. It's a 50-50 chance to which. If she bred to another immortal horse, it'd be another 50-50 chance of expression - so more likely the foal would express immortality, but not guaranteed, as the foal may just be rolled to carry immortality.
3. Her immortality is expressed regardless due to it being a joining trait, but you need to go to trait genetics (
http://beqanna.com/forum/forumdisplay.php?fid=74) to have it rolled to see how her future foal(s) may be affected.
Also, you are by no means required to have genetics done, or use them for breeding. You're welcome to breed her without having a teg rolled. In those cases, the fairies simply use a stat calculator to see whether the baby inherits traits shared by both parents, father's trait(s) only, mother's trait(s) only, or not traits.