Fey Beast Tamer (theme)
Your companion shares your healing surge total. Whenever an effect requires your fey beast companion to spend a healing surge, the surge is deducted from your total. Whenever you use your second wind, your companion also regains hit points equal to your healing surge value. At the end of a short rest, your fey beast companion regains all its hit points.
If you die or your companion drops below 1 hit point, it retreats back into the Feywild. If that occurs, you can use one of these two ways to call it back.
Minor Action: You lose a healing surge, and your companion appears in the nearest unoccupied space, with hit points equal to your healing surge value.
Short Rest or Extended Rest: At the end of the rest, you lose a healing surge, and your companion appears in the nearest unoccupied space, with full hit points.
To answer your questions:
It can be healed during combat by using your second wind, or by taking one of the theme utility powers.
If it is hurt but not below 1hp, it will automatically heal to full during a short rest.
If it drops to 0hp or less during combat, it disappears. You can resummon it during combat with a minor action; this costs you a healing surge. During an extended or short rest it will reappear automatically, again costing you a healing surge.
Animal Master (theme)
Your animal minion normally stays out of the way in combat, waiting by the outskirts of the fighting. You can direct it to help you by using distracting attack. If you’re not using your animal minion to harass a foe, you can assume that your pet is safely out of harm’s way and won’t attract any enemy attention. Distracting attack doesn’t normally provoke a response against your animal, since it moves in and out of the thick of the fighting quickly.
For all your questions, the answer is: It can't be healed and it doesn't come back, because it shouldn't ever be hurt in the first place. If the DM thinks it should be hurt by something, point out to him that the rules provide you no way of replacing it; if he really wants to off it, he should allow you to pick a new theme.
Beast Mastery (ranger class feature)
Healing Surges: Your beast companion has two healing surges. The beast's healing surge value is equal to a quarter of its maximum hit points, as normal.
Beast Companions and Healing
Your beast companion can spend healing surges as any character can, and it can receive the benefits of healing abilities, such as a cleric's Healing Word.
Second Wind: When you use your second wind, your beast companion can use second wind as well.
Other Healing: When you are adjacent to your beast companion, you can spend a minor action and one of your healing surges to heal the beast companion as if it had spent a healing surge.
Death and Dying: A beast companion follows the same rules as a character for death and dying... If your beast companion dies, you can use the Raise Beast Companion ritual to restore it to life.
To answer your questions:
It can be healed during combat the same way a player would be, and you can also spend a minor action and a healing surge while adjacent to it to heal it.
It heals during rests the same way a player would.
If it dies you can resurrect it with the Raise Beast Companion ritual, which you automatically know if you are a Beast Master ranger.