Keep dungeon quest active?

For ongoing access to dungeons, do I keep their related quests active, or are they unlock unpon completion of the quests?

"Watch it Jiggle" let's you into the Ooze pit. If I accept the reward, is the quest repeatable or does it permanently unlock the pit for later exploration?

After you accept the reward for a dungeon quest, it will disappear and you will need to get that same quest in order to gain access to the respective dungeon. And you can get it (again) from the same NPC that gave it to you the first time. So no, it won't be permanently unlocked for later exploration.

The quests can be repeated. Thank you.

Yes, all the dungeon quests can be repeated except for one, Heart of the Shiverpeaks, which is a primary quest and can be done only once.

If you have it, you will get Cyndr the Mountain Heart as the final boss (which has an unique mechanic in that you need to drop a keg nearby, multiple times, in order to make him not-invulnerable to damage that isn't life stealing or degeneration).
If you don't have it, then everything will be the same but, instead of Cyndr, you will have to defeat Magmus and a lot of Oozes. And you will get a chest and recognition for completing the Heart of the Shiverpeaks dungeon, just like before.

These dungeons I been through are open dungeons are as far as I know;

Oola's Lab *accept from Blimm to gain entry*

Shard's of Orr *From Rata Sum to Gadd;s Encampment*

The Crevasse *Outside of Lion's Arch in N. Kryta Prov.*

Battledepths *Outside of EoTN*

Cathedral of Flames *Accept from Gron Pieceheart in Doomlore Shrine*

Several others I been to that I can recall atm. But these should be interesting to explore as much as I like to explore.