Alot of your choices don't make much sense if im honest with you. So heres my opinion going from character to character.
--BTs to Come--
Yda Hext: Breaking Boundaries has zero reason to play for her, it never plays in any scene shes in, likely Yda's BT them would be Torn from the Heavens.
Lyse Hext: Liberty or Death: This one i agree with, since it plays during the chapter Lyse was brought in.
Papalymo Totolymo: Another Brick: Its hard to tell what Papalymo would get. But Bealsars Wall's theme would fit better here. (If your passed heavensward you know why.)
Thancred Waters: Torn From The Heavens is a good generic choice, but i would prefer he gets something else. But I can't think of anything else for him, only for Gunbreaker Thancred, which i would choose Insatiable for him since it plays when he fights Ran'jit.
Alphinaud Leveilleur: This is a hard disagree, this song has little association with Alphinaud. And i would probably give him one of the two Heavensward Dungeon Boss fights. Probably Omnious Prongosticks (I know its Y'sh's but im pretty sure they've used the same song a few times on other BT's) Revenge Twofold. I say this because his kit and design is taken from Heavensward.
Alisaie Leveilleur: This is a tough one. I actually like the song you chose and has been used with her. HOWEVER, i would probably think they'll give her the Binding Coil Boss theme from Coils 1 - 4 Calamity Unbound.
--Possible Future Units--
Urianger Augurelt: ARR's Boss theme would only make since if hes brought in as 1.0 Urianger, which would be dumb. He'll likely be brought in as Astrologian, so his theme would likely be from Shadowbringers or Endwalker. It would likely be Endwalkers Mid Boss theme or Final Dungeon boss theme.
G'raha Tia: Why? Where All Roads Leads? This song has little to do with G'raha in the end. Likely his song would either be Eternal Wind - Shadowbringers Version, that plays in 6.3 after the trial.
Estinein Wyrmblood: Revenge of the Horde: This is a good choice, No notes.
Krile Mayer Baldesion: Magicked Skulls: Primary battle theme for The Forbidden Land Eureka. Another good choice. No notes.
Minfillia Ward: This is a hard one. I like the Aetherial Sea song, but that song isn't exclusively about Minfillia, while she plays a role in the Dungeon, the song i would associate most with her, would be the Antitower's theme.
Moenbryda Wilfsunnwyn: Hard to Miss: FATE battle theme. Eh this would be good enough, sadly Moenbryda has no songs associated with her.
Loiusoux Leveilleur: From The Ashes: Boss theme for Phoenix, Loisoux's Primal form. Decent choice. No notes.
The Grand company leaders are good choices, i agree with all three of thesse. Elder Seedseer Kan-E Senna: The Forest's Pulse: Battle theme of the Black Shroud in 1.x
Chief Admiral Merlwyb Bloefhiswyn: Nail of the Heavens: Battle theme of La Noscea in 1.x
General Raubahn Aldynn: Quicksand: Battle theme of Thanalan in 1.x
Gaius van Baelsar: If they give him Ultima when he already technically has not one, but two battle themes asscoicated with him, i will yeet myself to the square office to smack some sense into them. Seriously, dude has two hole themes associated with him. Bite of the Black Wolf which plays when we fight him in ARR or The Black Wolf Stalks again (Which in my opinion they should pick this one, since the song comes from a sidequest series directly about Gaius). While Ultima would be a decent pick for him there are far, far better choices imo.
Nael van Darnus: Rise of the White Raven: Final Boss Theme of the 1.x storyline, as well as Nael's personal battle theme in all fights against her. This is litterally her theme, no notes.
Lahabrea: Thunderer: Battle theme of the Ascians. Given recent developments. Theres now more options for Lahabrea. Embers or White Stone Black from the newest raid series would also work, but they likely would give him thunderer.
Haurchefant Greystone: For The Sky: Cutscene and mission theme in Heavensward. Haurcefants a hard one but im gonna make a strange case here. Instead of the generic theme used for cutscenes and missions, Haurcefants theme should be Imagination. While he doesn't appear in this dungeon, this song always spoke to me, as the WoL wanting vengenance.
Ysayle Dangoulain: Oblivion: Theme of the second phases against Shiva, Ysayle's Primal form. This is the only logical choice for her.
Aymeric de Borel: Stone and Steel: Cutscene theme in Heavensward.I would have choose Faith in Her Fury, over Stone and Steel for him personally. But i do understand why you would say this for lucia.
Lucia Junius: Faith in Her Fury: Theme for the Steps of Faith trial.
Hilda Ware: Melt: Standard battle theme for Heavensward. No notes, its a good choice for her.
Yugiri Mistwalker: Triumph: Final dungeon boss theme for Stromblood.Good Choice for her, no notes.
Gosetsu Daito: Earth, Wind and Water: This song is not associated with him at all. His song IMO should be Phase two Tsukyomi, Lunacy. I think this would suit him well given how his and Yotsuyu's story goes up til that point.
Hein Rijin: Gates of the Moon: Theme of the Doma Castle dungeon in Stormblood. This is a logical choice for him, i'll agree with it.
Cirina Mol: Looping in the Deepest Fringes: Stormblood's standard battle theme. Can't argue with this.
Sadu Dotharl: I would assign Most Unworthy to Magnai not Sadu, since if fits with him alot better, the songs not chaotic enough for Sadu. I would likely just give her triumph.
Magnai Oronir: Most Unworthy.
Fordola Lupis: Song of Salt and Suffering: Daytime theme of The Lochs. This is a decent choice. Though, Meteor plays when we fight her a second time, Her song really will depend on where shes pulled from.
Yotsuyu goe Brutus: Wayward Daughter: Theme of the final phases in the fight against Tsukuyomi, Yotuyu's Primal form. Thisis the best choice for her.
Ryne: Insatiable: Final dungeon boss theme for Shadowbringer. This is the wrong choice. Her song shouldn't be anything but "Return to Oblivion" from Eden.
Ardbert: Who Brings Shadow: Theme of the first phase against Hades, as well as Shadowbringer's primary theme. This or Ultima would both work.
Lyna: To Fire and Sword: Theme of the Holminster Switch dungeon in Shadowbringers. I hope Lyna makes it in and this or the Crystariums Day theme is her song.
Gaia: Promises to Keep: Final Boss Theme of the Eden raids, played against Eden. Either this or FFXIV's version of The Extreme would work.
Vauthry: Insanity: Boss theme of Innocence. Vauthry will likely won't be added, but this only song that should be given to him.
Emet-Selch: Invincible: Second phase theme against Hades, the Final Boss of Shadowbringer's initial storyline. Emet actually has not one, not two, but three good choices. Invincible is a very good choice, but so is Who Brings Shadow and Mortal Instants (Amarout's Dungeon Theme). All three would be good choices.
Elidibus: To the Edge: Final Boss Theme of the Warrior of Light, the Final Boss of the conclusion to the Shadowbringers storyline. This is the correct choice.
Venat: Your Answer: Boss theme against Hydaelyn. This or Flow Together. No other choices
Hythlodaues: Finality: Final dungeon boss theme for Endwalker. Eeeh this or the Mid Dungeon boss theme.
Fandaniel: Endcaller: Boss theme against Zodiark. Yeah this the right choice.
Meteion: The Final Day: First phase theme against the Endsinger at the end of Endwalker. This or With Hearts Aligned.
Zenos viator Galvus: Depending on how he is represented; The Worm's Tail, the Final Boss Theme against Shinryu, Final Boss of Stormblood and Zenos' Primal form, if he is a Samurai, or Endwalker, the theme the plays when facing him at the end of Endwalker, if he is a Reaper. These are both the only resonable choices. Though Meteor would also work since that is used during the first fight with him. Though Endwalker for Reaper Zeons would work well.
Zero: Scream: Theme of the bosses in the Sixth and Seventh Circles of the Pandemonium raids. This is a really hard NOPE. The song has Zero (hehehe) to do with the character at all. Her theme would likely be an upcoming song from one of the patches, or "Zero's Domain - Cry and Sorrow"
Edda Blackbosom: Blackbosom: Boss theme of Edda Blackbosom. Fully agree.
Inspector Hildibrand Helidor Maximillian Manderville: Agent of Inquiry: Theme of Hildibrand. And this didn't need to be said.
You made some good choices in here, but some that made me scratch my head confused on why you suggested them. Your choice for Zero confuses me the most though.