No. After this point, you’ll receive a pet dungeon-specific reward. Each pet dungeon has their own currency, so you can’t use a currency from Deadmines in the Celestial Tournament.
You can do each pet dungeon once per week per account.
ETA: The only exception to this is the first time you run the dungeon. You can run it once on normal mode, which gave you the Ultimate Battle-Training Stone, and then again on challenge mode (no healing pets). I believe the Celestial Tournament is the only one without a normal mode.
No. There are currently five pet dungeons:
Wailing Caverns : This dungeon gives no currency, but gives a bag of pet supplies on completion called Damp Pet Supplies. There are three pets that have a chance from dropping from this bag-- Cavern Moccasin, Young Venomfang, and the Everliving Spore. The Everliving Spore is the rarest of the three.
You’ll also receive a pet for completing this dungeon on challenge mode called Son of Skum.
Deadmines: This dungeon gives a currency known as “Old Bottle Cap”, which you can exchange for three different pets: Foe Reaper 0.9, Pocket Cannon, and Tricorne.
You’ll also receive a pet for completing this dungeon on challenge mode: Mining Monkey
The Celestial Tournament (Timeless Isle): This dungeon also rewards currency called “Celestial Coin”, and can be exchanged for four different pets: Yu’la, Broodling of Yu’lon; Xu-Fu, Cub of Xuen; Zao, Calfling of Niuzao; and Chi-Chi, Hatchling of Chi-ji
There is no pet reward for completion of this dungeon.
Gnomeregan: This dungeon gives a currency called “Pristine Gizmo”, and is the currency for three pets: Alarm-O-Dog, Leper Rat, and Mechanical Cockroach.
You will also receive a pet for completing this dungeon on challenge mode: Mini Spider Tank
Stratholme: This is the newest dungeon, and rewards a currency called, “Cleansed Remains”, and is used to buy four pets: Gruesome Belcher, Shrieker, Ziggy, and Crypt Fiend.
This dungeon is unique, in that there are two special NPCs inside. One is Postmaster Malown, and beating him will reward a pet called Burnout. The other NPC is Fras Siabi and will reward an additional Cleansed Remain if you’re able to beat him.
This dungeon also gives a pet as a reward for beating it on challenge mode: Minimancer