Do companions die?

Solution 1:

Follower death is handled pretty oddly;

• If their health is exhausted due to enemy fire, they just get knocked out. Healing them or finishing combat and allowing them to regenerate will bring them back to life.
• If you kill them due to friendly fire, that's it: they're dead permanently.

But there are situations that may cause a follower to die permanently, even after they enter the "knocked out"/"cower" state: if an enemy keeps wailing on them, there's a chance they'll die permanently.

But it's not guaranteed: looking around online, I've found numerous accounts of Lydia or Faendal dying easily, but others who swear nothing they do kills them. UESP describes the follower death mechanics as such:

When your follower would die to an enemy attack or a trap, they will be incapacitated for some time and begin to recover their health, just like essential NPCs; enemies will stop attacking or targeting them during this time. However, if they lose all their health again during this period, they will die for good. Thus, you should try to move the fight away from them, so that they are not hit by a stray arrow or area-effect spell. They can also die if hit with a damage-over-time poison that is stronger than their health recovery. They can also die from enemy power-attacks.

Solution 2:

Followers can be killed sometimes.

From the Elder Scrolls Wiki, followers who can be killed can die from:

1. Too much damage (e.g., a giant can kill Lydia easily)
2. Being killed by the player(including shouting)
3. Falling
4. Poison
5. Traps
6. Being hit by an enemy while the player is speaking with them
7. Being attacked by turrets

That said, there are certain circumstances where followers are immune to death. For example, Mjoll the Lioness:

Mjoll is an essential NPC and cannot be killed by other NPCs or by the player.

On the other hand, Aela the Huntress is immune to death during a single quest:

She cannot be killed while the Totems of Hircine quest is active because of her quest significance. However, once the quest has been completed, she can be killed.

Solution 3:

Followers can die, you can even kill them yourselves.

They don't generally rush ahead, they attack if you attack or you are attacked. They won't do anything if you're not in combat. They will even sneak when you sneak. You can also command them to "wait" if you want to go it alone on a certain part.

They have a "second chance" of sorts - when they are about to die they fall to the ground and recover, just like important NPCs. The mobs will switch their attention to you. If the mobs go back to the follower after that, he will die tho.

Solution 4:

Companions can die, however most of the time when they are low on hitpoints they start to crawl and regenerate health. At this time the enemies will start attacking something else (usually you).