Keep in mind that these quests are optional, and are not necessary to continue leveling and capturing companions. They are also account-wide, so if you complete the quest chain on one character, you will not be able to complete it again on another. However, any progress that you've made will be saved and you can pick up where you left off in the quest chain on another character. The dailies that you've unlocked will also be accessible by all your toons.
One player said in a comment posted in response to the list, “Yeah people don’t realize the sheer enormity of game system evolution WoW has gone through since release. I’m not the biggest fan of BoA by any stretch, but I’ve played since closed beta vanilla, and I doubt I’ll be going back to classic. Leveling was painful. Experiencing these old systems once was enough.”
Analogy: think back on riding the swings in kindergarten. Was it fun? Now, imagine going back to those swings exactly as they were, and sitting in them now, as an adult. They're too small. They don't fit. Your feet drag on the ground because they're so low to the ground. The bar over your head is low enough that you can reach out and touch it, and even at the highest the swing will go, it's only about chest high when you're standing up. It's exactly the same swing, but riding on it now is a very different experience, yes?
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.
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.
2. Achieving. There are many pet battling achievements in game, and following the natural trail through the Taming [Continent] achievements up to Pandaria, and then on to the Spirit Tamers, Beasts of Fable and the Celestial Tournament was an epic journey for me and for many others. It's well worth doing. The Celestial Tournament is still, IMO, the pinnacle of PvE pet battling. However, WoD and Legion and BfA have added more battling achievements.
Through out the game, the quest log will be capped to 20 many times. The guide makes use of maximizing the total amount of quests you can accept at once. Because of this it is important to follow the guide exactly and only accept quests if the guide tells you to so you won't run into issues not being able to accept a quest. If you do accept a quest that is not listed in the guide, write it down because you may need to abandon it in order to keep following the guide if your quest log is full.