When I look at my horse, for example, the horseshoe icon or symbol appears meaning I can rid it, but sometimes this pale yellow horizontal curve appears just above it. what does that mean?

That's the NPC health bar.

It displays whenever you target something with less than full health.

How do I give my horse some health? Press T and wait for an hour? I'll try that.

Thanks for the answer; that's disturbing when it's my good ol' horse.

You can buy or make a restore health on target or touch spell to to heal your horse. You can buy the spells in Chapels (convalescence spells).

Just to note, Convalescence is horrifically inefficient, and you'll be out of magicka before you can restore any appreciable amount of health to your horse. I'd recommend making a custom healing spell at the Arcane University, once you've unlocked it through the Mages Guild. Remember to make it a healing spell on touch, not target. Target is partially what makes it so inefficient. Another thing to remember is to make the spell heal over time: 10 points over 2 seconds is more magicka-efficient than 20 points instantly.