Why the repeated mention of time travel in the description of the Te Oria and the lost technology? It sounds like a huge Chekhov's Gun in the offing - the Te Oria gets triggered to destroy the world, so they Take a Third Option to avoid having to use the solution they came up with.
So what was with Faris going berserk and almost randomly killing the little girl?
She got tunnel vision.
It makes sense that the Big Bad arranged for the doctor and his two minions to get into the party, but wouldn't someone notice that the guy who attempted a coup d'état just a bit back was trying out h'ors d'oeuvres?
I think he had a monocle, so he was effectively anonymous.
If all they needed was for two people to have a near-experience with death, why not just fall off the cliff again and be saved by someone on the ground?
They seemed quite comfortable with the current situation in the end...
Near death experiences have an annoying tendency to become actual death experiences.
And if they knew it was safe, it would defeat the whole purpose, as their souls wouldn't jump ship.
When Falis ends up in Alita's body, why is she able to perform the exact same feats of badassery as before? Alita's body would have been significantly weaker than Falis'. Hell, Alita was exhausted just on the way from the castle to the forest. Yet as soon as Falis ends up in Alita's body, she's jumping around like an Olympic gymnast on speed. I know, I know... Rule of Cool. But still.
Alita's body may have a few secrets of its own, like how if overstressed, it can enter a berserk state. Falis states it never happened to her in her old body. And those natural red eyes aren't a common color at all, at least among humans.
Falis hates the corset. Hates it, hates it, hates it. So why does she keep wearing it in the last two episodes, when she doesn't need to? (Changing clothes entirely would've been smart for several reasons, but it's the corset that really sticks out.)
Alita's "Milano" disguise poses problems in both versions.
In the manga, we don't meet any real Milano; Alita just says she's now Milano Entrasia, Jodo's granddaughter. How could that possibly hold up to scrutiny? Either there's no real Milano (in which case Jodo suddenly has a granddaughter out of nowhere) or there is (in which case someone will inevitably notice she's in two places at once). Why not make up the whole identity?
In the anime, there's a real Milano Entrasia whose persona Alita takes on. But Only Six Faces isn't supposed to be true in-story — everyone who knew Milano should immediately see the difference. How can this work without at least suspiciously firing all the maid staff?
She takes on the name Milano to honor her memory. Milano acted as a decoy to allow Alita to escape and lost her life for it. So it's less about making cover and more about cherishing a fallen friend. The castle staff was probably told some cover story: the Princess wanted a Replacement Goldfish, Jodo has a second grand-daughter named "Milano", one of Milano's friends (whom they've never met) wanted to honor her memory (kinda-sorta the truth) etc.
Related to the above: in the anime, Milano was Jodo's real granddaughter. We see Alita mourn her, but why doesn't he seem affected at all? Losing a granddaughter and having to pretend she's still alive is not the kind of thing you take in stride! (And if they didn't have time to show it, why bother making Milano related to him?)
While we know nothing of Jodo's personal history, it is possible he had moment of solace off screen or it's not his nature to mourn.