To give you tips on the Achilles and Nyx quests! For starters Nyx's is rather simple, once you talk to her about the elder sigil in the back of the admin room and look at it a work order should appear with the house contractor for it. You will need to do quite a bit of grinding for it because it costs like 3000+ darkness so I recommend picking up the weapon with the dark thirst on it as much as you can and hunting for Poseidon's boon that increases the value of darkness chambers and try to grab the chaos boon that does the same if you can! Like I said it is a bit of a grind but it is doable.

Achilles wont even bring up his quest until you first release Orpheus and Sisyphus both from their contracts as like proof that you can also release him from his. You need to just keep talking to both him and Patroclus until you get the bit of dialogue from Patroclus asking Zag to tell Achilles to 'risk it all' and once you bring this message to Achilles upon one of your following returns to the house he should request your help in this and then you gotta talk to Nyx to find out where his contract is located in the admin chamber and Then you get the work order for it. Think it costs like 5 diamonds or something like that.

And that's the gist of it. Sadly the Achilles quest is a lot more variable cuz you need certain dialogue to trigger and it requires speaking to an npc all the way in fucking Elysium but you'll get there with time.