There are two parts of the quest chain for each continent. The first part involves defeating numerous Master Pet Tamers, and the second requires you to triumph over the Grand Master Tamer. After you've bested the Grand Master Tamer, dailies from each Tamer will unlock for that continent. All the dailies continue to reward experience for your companions, however only the Grand Master daily quest will reward the Sack of Pet Supplies.
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.
Most people come here not to read walls of text, they look for the better-rated strategies that match their pet collections, copy&paste stuff they need to import into their addons and only come back if something goes horribly wrong and then mostly to find help in other peoples' comments who also struggled or to find a different team/strategy: I don't think many people wanna read about stuff that already happened to them, they mostly want to see solutions and it's the best way to have them in a well-optimized strategy&team already.