Lydia was nothing but wormfood for me, died constantly and forced many a reload of the game, had to leave her at home the very second i bought my Whiterun home and finally had a chest to store my excess loot in (acting as a mobile wastebin was basicly the only use i had for Lydia anyway).
Companions can absolutely be killed, once their health reaches 0, they go into this "kneeling and out of breath" sort of animation, in this state they will regen health, but if they sustain more damage in this state, they can also be perma-killed by enemies.
Thing is, i decided to be an archer from the start (archery seems one of the least shallow combat styles avalible, along with mage which i'll probably try in my next playthrough), but the companion AI doesen't deal well with that.
To do damage with a bow i obviously want to keep my distance to the enemies, that's my one and only advantage with a bow, but the companion AI seems hardcoded to not be impressed by that, indeed, once you get clouse enough to an enemy to hit them with an arrow, you are also clouse enough for the companion AI to enter the "mindlessly run over to the baddies and commence hack'n'slash" routines of their programming, which means they will run infront of you and compleately block your every attempt to hit the enemies, or even help them as they are getting pwned, because they are in the way and blocking your every shot!
Ahh, but once they get pwned and fall to their knee's, they are usually safe enough... yes, unless there is an enemy archer or mage in the bunch, because they will then shoot at you, but since your companion has brilliantly decided to situate him/herself directly betweeen you and the enemies, that means they will allmost certainly kill the companion first as they try hitting you thru them!
*groans*
I wish i could say this was just an oversight made in Skyrim, but the exact same problems existed in Fallout3 and New Vegas, and is why i never ever used companions in thouse games either, all they ever do is run infront of you and block your damned shots.