Skyrim:Hard Answers

Quick Walkthrough
#Speak to Enthir at The Frozen Hearth or Hall of Attainment.
#Speak to Calcelmo in Understone Keep.
#Enter the Dwemer Museum.
#Copy the Falmer Translation Guide.
#Bring the copy of the Guide to Enthir.
#Speak to Karliah in The Frozen Hearth.

Detailed Walkthrough
First go speak with Enthir at The Frozen Hearth in Winterhold. After speaking with Enthir, you'll be directed to the wizard, Calcelmo, at Understone Keep in Markarth.

: Note: There is a bug that dictates you cannot talk to Enthir if you have started, but not completed the quest The Staff of Magnus, as Enthir believes Winterhold to be in danger and will continually repeat "This is going to be terrible for business." 

Calcelmo, the Wizard
You can find Calcelmo overseeing the excavation site in Understone Keep. He's not happy to be disturbed and no matter what you choose to do, will not be interested in helping you translate the journal - at best, he will grant you access to his museum. You have several options on how to proceed:
You can go straight to the museum and pick the lock.
Using this method causes the guards in the museum to be hostile, and as they are city guards you will incur a bounty if you kill them.
You can simply steal his key, which should be right next to him on the table.
This option will cause the guards to be hostile, unless you are successfully hidden.
If you ask to visit the archaeological site, you will be asked to kill Nimhe (a giant frostbite spider). After you have done this task you will be given the key to the museum, and can walk past the city guards without sneaking.
Solving Calcelmo's love problems during The Book of Love will add additional dialogue where he will hand over the key to the museum, and allow you passage (the guard at the door will still tell you you can't enter, but the guards in the museum won't bother you).
If you can pass a hard Speech check, you can persuade him to give you the key to the museum.


Enter the Museum
Whatever you chose to do, it's clear you're going to have to break into Calcelmo's laboratory in order to retrieve the Falmer Translation Guide, so head for the Dwemer Museum. There are a lot of random Dwemer artifacts and books in the museum, but most notably, there is a gate on the south side protecting a room containing an Unusual Gem - the gem is located on a table to your right as you enter the first room on the left. Make your way to the far end of the museum to find the entrance to Calcelmo's Laboratory. As you enter the lab, you can find the Spider Control Rod in the room to the left, which you can use to help clear out the guards ahead (note that the rod only works in the lab). 

There are two types of guards in the museum and laboratory: Markarth City guards and Calcelmo's personal guards. The city guards here will not be hostile (provided you didn't pick the Museum lock), but Calcelmo's personal guards in the lab will confront you if they spot you, no matter what (though they will not give you a bounty if they are harmed). It's possible to sneak past all the guards in the lab without any combat, but you can also employ the various traps around the lab to dispatch them quickly. Whatever you choose, make your way to the end of the laboratory.

As you make your way through, beware the hall with the poison gas; your health will drain rapidly until you reach the other side, and it is not possible to prevent the damage with poison resistances, even with items such as Hevnoraak. Putting 40 pounds of weight (either the body of a dead guard or a heavy item such as Dwarven armor) on the pressure plate at either end of the hall will depress the plate and stop the gas from spreading, allowing you to comfortably loot the corpses. You can also Become Ethereal and not suffer the poison for the duration of the shout (though be warned, this may alert nearby guards). 

At the end of the lab, you'll find Aicantar's study, where Calcelmo's nephew is busy with his research. Activating the trap here will send Aicantar into a panic and he and his guards will make a run for it (or you can just sneak past him). Near the exit, you can find the Dwemer Puzzle Cube for The Litany of Larceny quest. Exit out onto the Wizard's Balcony and head up the stairs to the right into Calcelmo's Tower. In the tower, follow the hall to the left to find a room with a desk. Beyond the room is the Falmer Translation Guide, which turns out to be a massive stone tablet. There's no way you're bringing the whole thing back to Enthir, so grab a piece of charcoal and paper from the previous room and use them to create a rubbing of the tablet.

Return to Enthir
After you copy the text, two Wizards' Guards, Captain Aquilius, and Aicantar will enter the tower to hunt you down. You may fight them if you wish, but you can also simply sneak past them after they split up. Distracting the remaining guard with an arrow is a good way to aid a stealthy approach. For the quickest escape, look towards the exit from the tablet and you'll notice a small ledge on either side of the wall over the door. Move from behind the tablet onto the platform to your left, then jump onto the little "chimney" at the end of the platform. From there, you can jump onto the small ledge and proceed directly to the door.

Once outside, a waypoint will indicate your exit route: simply hug the right wall of the balcony to find a gap leading to a small cliffside path, terminating at a waterfall. Carefully jump into the pool at the base of the waterfall to complete your escape. It's also possible to "climb" down from the balcony by looking left and down when you reach the waterfall - there will be a path of sorts where you can make very small drops to multiple rocks (you can use Become Ethereal) to avoid damage with either method). Alternatively, you can also just retrace your steps back through the museum and exit the keep the same way you came in.

Once you've escaped, return to Winterhold, where Karliah and Enthir are waiting in the cellar of the Frozen Hearth. Enthir will use the rubbing of Calcelmo's tablet to translate Gallus' journal, confirming that Mercer used the Guild's funds and desecrated the Twilight Sepulcher, the secret shrine to Nocturnal that the Nightingales are sworn to protect. Karliah will also give you the Nightingale Blade as a token of appreciation, and ask that you meet her at the Ragged Flagon. After talking with Enthir and Karliah, the quest ends and The Pursuit begins.

: Note: Completing this quest will lock the backdoor from Riften's graveyard to The Ragged Flagon - Cistern until later in the next quest.

Guild Trinket
The trinket in this quest is the Dwemer Puzzle Cube. It can be found in the Calcelmo's Laboratory zone, in the room where Aicantar is reading. The Cube is on a table to the right, after going past Aicantar's table but before the door to the balcony. Once you've collected the trinket, it can be returned to Delvin Mallory for some leveled gold.

Notes
The highest level version of the Nightingale Blade is untemperable (levels 46+), making the level 36-45 blade the best version. However, all versions lack a smithing perk, so they will quickly fall behind other swords when you start tempering.  
If you kill Aicantar prior to copying the Falmer text, he will not arrive with the guards and the city captain. Instead, there will be one extra guard in his place.
Dawnguard reveals at least one NPC who could conceivably translate the Falmer language, but doesn't add a way to ask him to do so.
When you take Gallus's Encoded Journal to Enthir, if you ask about the Falmer he will give you a brief history and then say that he has a book on his table if you are interested in knowing more.  As he is found in the inn during this quest, this may be a reference to The Falmer: A Study found on a table in the Frozen Hearth Cellar.

Bugs
 
 
 
 Enter the console command   after completing the quest.
 
Another possible solution is to leave and wait for normal store hours and Enthir becoming open for business.
Sometimes leaving the building and reentering will get his correct dialogue running. His dialogue changing to a store (fence) shows his script has been completed and he is out of the picture for this quest.
 
A possible workaround is to sneak into the room with the guard and use a bow to kill him. Even if he is running towards you, it seems if you can kill him before he starts his formal dialogue, the bounty is not added and you can play on as normal.
 
This seems to have to do with having Mercer's plans in your inventory. This is an important quest item for the next quest and the quest fails to load as long as this item is in inventory. You should drop the plans somewhere, and retrieve them when you want to finish The Pursuit. Use command Startquest TG07 then Set stage TG07 0 to start and add the quest to the journal while keeping Mercer's plan, when they're "quest items" and it doesn't allow you to get rid of them.
 
Do not complete this quest a second time. Doing so will break the end of the "Darkness Returns" quest; Karliah will not let you out of the last room and will only say "Yes, fellow Nightingale?" when spoken to. You will be stuck in that room permanently. The only fix for this is to reload a save from before you started this quest a second time. If you do this after Darkness Returns, you will be able to keep both Nightingale Blades. 

Quest Stages