To do the Halls of Testing quests in Temple of Veeshan you’ll need Ally faction with Claws of Veeshan.
Note: You may be able to do the quests at warmly faction but I have not personally tested it. You can speak to the dragons in the West Wing and see if they will give you quest dialogue. If they give you the quest then your faction is sufficient.
You’ll kill Dozekar the Cursed and and mini drake bosses in the East Wing (AKA Halls of Testing) to loot quest items.
Neither Dozekar the Cursed or the Named Drakes drop the same
You can hand these quest items to the specific dragons (see details below) in the West Wing: Gozzrem, Lendiniara the Keeper, or Telkorenar to receive some of the best loot in Scars of Velious.
Respawn timer in Temple of Veeshan is 3 days (with variance) for all dragons involved with these quests.