This is something that people really need to start to understand, not just for Classic or WoW but software developement in general. Fixing a bug isn't just flipping a switch or changing a single digit. You have a machine with literally hundreds of thousands of parts, designed by dozens of different people at different times over the course of 20 and more years. You change a single thing and it might break 50 other things in very subtle ways that might not even emerge until months later (i.e. the very high M+ keys some players got at the start of BfA).
One thing they told me when I dabbled into computer science back at university was that 80% of the bugs only take 20% of your time to fix. That's why sometimes you see issues persisting for very long times, not because the devs are lazy or incompetent, but because there is usually a priority list of problems, with the severe-but-easy-to-fix ones always taking precedent and the not-game-breaking-but-difficult-to-solve problems being last line.
It is completely acceptable, but harder to write, that your character has been a demon hunter sometime after the War of the Ancients. This is no different than people who have written ancient druids, and it poses the same exact problems in writing. I would not recommend this to anyone who is not a seasoned veteran when it comes to writing characters, as the longer your character has been a demon hunter, the older your character is, the more difficult to control the fel magic within, the more detached from society, and the more withdrawn they may become. Overall, it becomes incredibly harder to write.
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.
A new quest will pop up once your first pet reaches level 3. It will send you to either Audrey Burnhep in Stormwind or Varzok in Orgrimmar. These trainers will offer a quest chain to battle specific Pet Tamer NPCs on each continent. Defeating each of these NPCs will reward a nice chunk of experience for your pets, as well a Sack of Pet Supplies (which has the chance to drop the Porcupette companion, among the always-useful pet bandages).