There’s only a few pets that can heal themselves through passives, but you can’t heal them yourself without pet potions. If it dies, you can just re-summon it, but it will lose loyalty. (When it reaches 5 loyalty, you can no longer summon it). And yes, mounts do not have health.
The loyalty thing, what does it even mean? Can pets run away? I noticed that sometimes, a pet stays behind and gets out of sight, usually when I’m attacked while traveling. I go on, petty fights back and I lose it. So far, I recall it then, and re-summon when necessary. Does this have anything to do with loyalty?