How do I care for my horse's well being?

He can die, but it is like the NPC's it is quite hard unless you take a big fall, or you kill it yourself. Other can kill it but it requires that they try to hit you but miss, and then hit the horse in "rest" mode. In "rest" mode the bad guys will ignore the horse so just make sure that you are not close to it.

Shadowmere cant die.

And you can use the spell wich heals allys to heal your horse if you like to.

You dont need to feed your horse.

I havent seen any possibility to see your steeds health.

You can see his/her's heath if you hit it yourself, just be sure to hit him/her with your bare fists.