If a companion dies during the battle, it will need to be resurrected before it can participate in another encounter or be summoned as a traveling companion. Get that pet a medic! Or in this case, a Stable Master. Stable Master NPCs can heal and revive your pets for a small fee, and there is no cooldown on the Stable Master's heal. You’ll find Stable Masters in major towns and cities across Azeroth.
Only really to expand on Gráinne's comment as regards the 'collecting' and 'roster development' sections - which are, as said 'distinct' although they are intrinsically linked - I would like to point you in the direction of the in-game Achievements > Battle Pets > Collect tab. There are a series of achievements called Raiding With Leashes, pretty much all of which can be completed via the auction house - if you have the gold/inclination - or by completing old raids.
You’ll get a small window inside the game with instructions for what you need to do next, every step of the way to 120. You’ll also get a waypoint arrow that points you in the right direction to go at all times. If this sounds extremely convenient, you’re right! Zygor’s leveling guide/addon has completely taken the frustration out of leveling for me, personally. No more reading comments on WoWHead about how to complete quests. No more getting stuck, frustrated and bored with leveling. And no more feeling like WoW takes too much effort.