Can you lose a purchased horse in Skyrim?

If you buy a horse in Skyrim, is it possible to lose it?

If so, what are the scenarios for loss? (can it killed? If you leave it somewhere and lose it, do you have to go find it, will it ever "despawn", etc).

I'm trying to decide if it's worth the 1000 gold.

The horse can't "disappear" permanently, just like your companions can't. If you somehow lose track of your horse, you simply need to fast travel somewhere, and the horse will show up there, too.

If you fast travel inside a city, the horse will usually be teleported to a stable outside the city or to some other location around there if the city doesn't have one.

However, horses in this game are particularly proud (they're nord horses after all), so what might happen is that your horse runs away, gets attacked by something and tries to defend itself, dying in the process. Sadly, the "fast travel" trick won't work if the horse is dead.

There is also a random stray horse, free to take without stealing in between Valtheim Towers and Ivarstead

Horses are worth it as it allows you to carry more stuff back to sell. Load yourself up until you can't run then stagger to your horse. Their a pain as you have to get off to fight or even just cast a spell. As others say fast travel will usually reunite you with your steed if they are still alive. They do like to fight but only if you get off them. They can cover steeper terrain than you on your own and can outrun bears and most other dangers. If you fall from a hight your horse will die but you will probably survive this can be a useful escape if you haven't saved in a long time.

I'm not sure about if it can be "lost" (I assume it would just teleport to you when you enter a new area and it was lost, much like NPCs).

But yes, it can be killed. And horses... tend to attack enemies.