Xu-Fu can scan this section for the most suitable strategies according to your preferences and show the required pets.
Depending on the amount of fights in the section, this can take up to a minute to process.
Can you clarify which you mean by "recommended pet lists will be available again?" The only thing remotely like a pet list that has ever disappeared is the Celestial Tournament Planner, and that won't be coming back - certainly not in the forseeable future. It was based on an older version of the site's code, and couldn't reasonably be brought forward. However, the CT is so much easier now, with all the extra pets, that most people won't have to tune their roster so tightly for it anymore.
This list of the Most Used Pets is excellent, as a recommendation for top-level battles in general:
and I have a comment at the bottom about preferred breeds for those species.
In one sense, it is very important to have these pets. They are, for the most part, the "best" pets. However, more specifically, they are the pets most people use for making strategies, which means they are the ones you will see most when copying a strategy from here and other sources. In another sense, it is not important - except for the Family achievements, the battles can be beaten in many, many ways, and I'm sure they could all be completed without using any of those pets, but it would be harder, and you would have to design your own strategy yourself.
Edit: I forgot about the Legion Ready-Checker, which was a bit like the Celectioal Tournament Planner. There is an item in the Devlog about developing a new one, but no date specified. https://wow-petguide.com/?m=DevLog(edited)
I haven’t done the 10 PvP battles given to me from the guy in Org. Is that why I can’t seem to fight against these trainers? I have done the quest to kill certain Tamers in Pandaland, but still can’t fight the elemental ones.
I've done all the quest and i still can't fight them there is a daily quest called "What We've Been training For" do i also need to do this quest i got the title Tamer and Achievements Taming Pandaria, Taming Azeroth and I Choose You all at once after i defeated Master Aki but i have no more quest?
Never mind i figured it out i went back to Stormwind when i should of gone to shrine of seven stars gives me a quest called Beasts of Fable
You will have to complete the pet quest chains first. Check your quest log for any pet related quests. If you don't have any, perhaps check the pet trainer in your capital (Orgrimmar or Stormwind) to pick up the next quest. Hope this helps!