On the issue of all the timey-wimey stuff works, follow the scenes with Lutece Twins and the backstory surrouding them:
The first time you meet them right after arriving in Columbia - they ask Booker to guess on the result of a choin toss, after you did and the coin came up heads(?) they turn around walking away bantering about probalibity and you can acutally see a score of that coin toss experiement, which is like 100 to 1.
The second time is when the approach Elizabeth and gift her one of two broches, which you have to pick. The choice appears to loaded with symbolism but is ultimately inconsequenctial.
Both scenes confirm that the game is runing on version of a deterministic cosmology, despite having a multiverse ("There is always a lighthouse" or something like that). However, as said before, determinism is not one of action and word but only concerns character traits and subsquent patterns of behavior of principal characters. The situations way vary when making the side-ways steps through universes but the people are always the same.
As for the temporal sequence:
When you examine the Luteces' lab you get backstory of how they are actually two of them - as originally there was only one but in the investigation of Elizabeth's power (which only exists because she doesn't belong into the universe she is in - "The univese doesn't like it if you mix it's [something]"). In order to explore the mechanics of that ability Lutece made contact with an alternate version of herself, which happened to male. However because their experiments were sabotaged they both got "removed" from original universes, and stand apart from normal conceptions of time and space. Which is also why they seeming only ever interact with Booker and Elizabeth during the game.
Their removal from a "normal" space-time-framework is a fact which you can find them debating as they appear to be digging (up) their own graves in the Columbia Cemetary - which at the time appears kinda non-sensical as they are debating the finer points of grammatically expressing their situation, i.e. in which tense to express their observations of what "now" for the player.
With that in mind you can actually see that continuum of the Elizabeths might actually be in same situation as the Luteces by the end of the game - that's also what explains their sudden insight into their situation. And if you look closely at the end, the Elizabeths don't blink out of existence at the same time but individually. To me that always suggested that basically all Elizabeth's are becoming a single transcendant entity, but there is also the interpretation that they just disperse like a normal crowd, but across universes and not across space in one.
re Who are the "real" Elizabeth and Booker:
Through the application of multiverse theory here all versions of Booker and Elizabeth are real in their own context. There is no "true" singular timeline or universe but all variations are equally real. The determinism the game establishes as a rule doesn't change that, as the fixed qualites are content of everyone's characters not their actions or statements: All vairations of Booker/Comstock act out the same patterns of behavior because despite variations in the situation, they are the same people.
Now that opens up a big questions regarding when actually said content of character was fixed - as the game in its text doesn't provide an answer, as the continuum of Elizabeths chooses the point in time to kill Booker/Comstock through a consequentialist method - going only so far back as to eliminate all possible timelines in which Booker/Comstock and any Elizabeth would interact - rather than making a decision on principle that e.g. would required them to prevent Booker/Comstock even being born.
But then again - the story is self-contained and internally consistent without resolving that question.
As for Burial at Sea
, re Sally:
Sally is that universes version of Booker's penance by proxy, that timelines "Bring us the girl, wipe away the debt." - it's basically how that Elizabeth confirms that Rapture-Booker is actually a Booker.
That he takes that case and how Episode 1 plays out affirms that that Booker is just like Comstock and the Booker that rescues Elizabeth in Columbia - and his escape to Rapture, a consequence-free, non-judgemental envoirment, did nothing to change that - actually it highly consistent with how Comstock/Booker try to live a world in which they are not accountable for their own past actions.
re Rapture-Elizabeth: (Tinfoil Hat Level of speculation incoming)
Note that Episode 2 never confirms a timeline for or affirms the reality of the scenes in Paris - that the Paris scene immideatly descents into nightmare suggests that it is some kind of fantasy or dream.
As such it is unclear if the Elizabeth that tracks Booker to Rapture comes from a time after the continuum of Elizabeths "attains transcendence". It may very well be that she is version of Elizabeth that was never rescued by a Booker (her tear powers however suggest that she was sold to a Comstock and transfered across dimensions), but then through her powers learnt that this had/would happen to another Elizabeth but not to her. This would also explain why she is so resentful of Rapture-Booker (because see above).
And also, to re-interate the point about "multiverse but with determinism" the transplantation of all character to Rapture and into what are decades into their own future, is actually inconsequential - Booker will be Booker regardless of situation because his character is the fixed quantity.