Suggestion: Buy a stack of Pickaxes from shops, to save you the trip of returning to Bastok to purchase more; pickaxes can break when mining and Mine Gravel (you need one) is a high drop, but not guaranteed, per mine attempt. You can also buy Mine Gravel (AH: Others > Misc), but you have to travel to Palborough either way, so it is not worth it usually.
If you are low level, you may want to bring Silent Oils to prevent aggro from the level 21-23 Quadav on the 3rd floor.
Locate a Mythril Seam, (F-7) on the first floor or (I-9) on the third floor, and mine for Mine Gravel by trading a Pickaxe to the point.
It may take several attempts at the same Seam to obtain Mine Gravel.
If you receive the message "You find a chunk of mine gravel, but your pickaxe breaks in the process," you do successfully get the mine gravel.
At (I-8) take the elevator to the third floor. At (I-6) you find a refiner. Trade Mine Gravel to Refiner Lid. Pull the lever on the refiner to activate it. Drop off the ledge to your left to the floor below you. Pull the second lever.Mythril Sand will be put into your inventory automatically.
You can go back to Bastok Mines quickly by taking the boat at H-8 on the 3rd floor.
If she gives you Trust: Kupipi, you have to zone out and back in to trigger the 2-3 mission cutscene. If you talk to her without zoning, she just talks about Nanaa Mihgo instead.
Everyone in your party must have this item to enter the BC, or have completed 2-3 previously in their current nation of allegiance. One person cannot get others into the fight who do not have the Dark Key, even if they are currently on this part of the mission, unless they have completed the mission in their current nation of allegiance.
If you are solo, make sure you are a minimum of level 30 and on a very strong job, such as a Monk using Hundred Fists. Kill the dragon before killing Searcher, or the dragon will kill you rather easily after Searcher has weakened you. Searcher does not melee you unless you are close to it or it has less than 50% of its HP, so it is relatively inactive if you focus on the dragon first.
Once the Black Dragon is defeated, another cutscene will happen.
A glitch may prevent this cutscene from occurring. If this happens, you can view the cutscene here (begin at 8:53). Alternatively, you can exit the dais and reenter from the main entrance to watch from the Goblin Footprint.
If you are solo, make sure you are a minimum of level 30 and on a very strong job, such as a Monk using Hundred Fists. A solo hero with 4 trusts should have little difficulty clearing this BCNM below level 30.
A glitch can prevent the victory cutscene from occuring. If that happens, you can watch the cutscene here:  (begin watching from 1:53), or you can exit the shrine and reenter from the main entrance to watch from the Goblin Footprint.
After you've won the BCNM, you can exit back into Palborough Mines, travel a short distance on that map to H-8, where you can board one of the lever-activated ferries and return to Zeruhn Mines (which is connected to Bastok Mines).