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).
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your character is, the more difficult to control the fel magic within, the more detached from society, and the more withdrawn they may become.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.
The main topic for Classic at the moment would be whether the game is actually harder or just has more redundant activities you have to do before doing the stuff that's really fun. This is actually a good discussion to have, and not just for WoW, but gaming in general. The main sticking point for now, as most people are on (relatively) low levels is the breaks between killing mobs, especially for mana users. In the beginning it's actually refreshing having to think about what you can and can't pull, not just running into a bunch of mobs and killing them in *insert current optimal time to kill Blizzard determined is the most fun*. The food/drink breaks after a few (or even one) mobs provide time to actually look around the scenery and the mobs you're facing, and while there isn't much strategy involved at these levels, you still at least pretend to plan out how you're going to get to that quest mob without being killed by the 4 random ones surrounding them. Not being able to pull whatever mob you want, and actually having to check what mobs are there when you don't have interrupts yet (casters tend to be a nightmare) is definitely more difficult in a real way than what we have today. As a caster, having to actually think about which spells to use based on mana cost (and perhaps even using lower ranked ones) is definitely more difficult and requires more engagement with your character than we have in modern WoW.