Different qualities aren't limited only to wild pets—many rare World drops and special non-wild companions have had their stats boosted. For example, the world-drop dragon whelps (Azure Whelpling, Crimson Whelpling, Dark Whelpling, and Emerald Whelpling) are of [rare] quality. Check your Pet Journal to see the color-coded rarity of your non-wild pets.
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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.
The quality of captured wild pets is random. As of patch 5.1, upon entering a battle the Pet Battle UI shows the quality of a wild pet (idicated by a color-coded border around its icon). If you're farming a specific quality, if you forfeit the match your team will incur a small damage penalty (roughly 10%). There is no accurate way to determine a pet's quality prior to the start of a battle.
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.