Bioshock Infinite: What doesn't make sense

The way I like to look at it is that the entire game, the entire trip to Columbia is just an allegory that depicts Booker's descent into madness as a result of the guilt he felt for the events at Wounded Knee and for selling his daughter to some loanshark to rid himself of debt. In this case the ending is nothing more than his mental state in the final moment before committing suicide.
In this variation, the Letuces are visualized by Booker as omnipresent simply because he was unable to rid himself of the debt in real life and Comstock is pretty much Booker's guilt manifested. He is racist and extremely overprotective of Elizabeth both traits resulting form the events and Wounded Knee (slaughtering indians) and from the fact that he sold his daughter. If you want to believe this version of the game, think of it in the same sense as "Life of Pi" or the final episodes of "Evangelion".

Of course, you can devise a way to make everything work even if you believe the entire trip to Columbia is actually real, because of the Letuces. If you think of the Letuces (there was probably only one Letuce in reality but he just chooses to appear as both a man and a woman) as a Schrödinger's Cat type of existence (both dead and alive, both present and not present) than it is actually possible within the laws of quantum mechanics for Letuce to interfere in multiple universes in order to create a sequence of events that would lead to it's own creation and it this way to create an infinite loop without a beginning, a loop in witch Booker is trapped. In this way, the ending is actually the way Booker breaks himself free of the loop, by accepting a fate in witch all of the universes where he became The Prophet and where Columbia was built to disappear. He is able to do that because by the end, Elizabeth is also a Schrödinger's Cat type of existence, and, again, according to the laws of quantum mechanics, it is possible for Elizabeth to interfere in multiple universes and to create a series of events where she was never actually born in the first place.

I personally like the first interpretation more, even if it is prolly not what Ken Lavine had in mind

The way I understood after 2 playthroughs and getting every voxphone is like this. The luteces are not omniscient, they are however omnipresent. Through the "accident" caused by Fink they got stuck between tears (dimensions) and yes they can see the tears and move through them but only the ones they had traveled before and not to anyone they wished. By moving along these tears they eventually came to the universe where Elizabeth invades New York and realize that what they previously did (help Comstock get Anna) actually destroys the world so they set out to fix things. You also have to see that they have tried to fix things 122 times before (as evidenced by the coin flip) and how they did it those times remains unknown (although I believe everything went same way except for Songbird always killing Booker, like old Elizabeth tells him), you never see those other times except for the one where you end up ditching the whole Elizabeth thing in favor of joining the Vox, which makes him a martyr but doesn't accomplish anything in the grand scheme of things.

What I never understood was this notion that the Luteces are indeed the same person. I always thought they were brothers. To me it made sense that Rosalind indeed had a brother that died at a young age and her studies into tears suggested she could bring him to this world to stay with her. That was what motivated her to appear to Booker (as the archangel) to suggest him the creation of Columbia and to contact Rosalind and further her studies into the matter. This would explain the money for the tear funding and experiments where Comstock participated and became sterile.

When Comstock became sterile he had a huge problem because he had already announced his prophecy about the heir of the prophet and had married Lady Comstock. Because of this he remembered Anna from his old life, but it was likely that he couldn't reach to her in that timeline (she would have been older and not be able to pass as a baby). So he made the Luteces go to the time in his life before the baptism to offer Booker the deal (baby in exchange for the debt). As Booker tries to get Anna back part of her finger stays in his universe and the rest goes to Comstock's universe. This is what gives Elizabeth her power to open tears at will (like she mentions she could do as little, imagine as a baby). The Luteces realize this, as well as Comstock and device the siphon, the tower and Songbird to use Elizabeth's energy in order to advance Columbia's technology by opening tears into the future. They do not control this power fully though as evidenced by the tears all over Columbia (and the ones Elizabeth can open) but neither does Elizabeth, she can open the ones that emerge but not open ones at will or to wherever she wants. When Booker does rescue her from the tower, he is taking her away from the siphon, which in turn allows her to regain some of her power, she keeps getting stronger until she is able to not only bring stuff from other dimension but to travel with Booker to them. She still cannot control everything in them at will and that is why you see the whole problem with Chen Lin and the dead guards, she is trying to combine the universes but is not powerful enough to do so without consequence.

The Luteces have seen this things but do not know whether or not the choices made by both Booker and Elizabeth will yield results in the long run (often talked about in the constants and variables dialogs) and that is why they appear sometimes to give a little "push" if you will in the direction they wish. Up to this point everything is indicative of another failure by the Luteces to stop the prophecy from being fulfilled. Now Comstock also knows about these events because he used Rosalind's device after they were "killed" but given how stupid he is (after all he is Booker), he has no idea how to correctly interpret things and believe things to be prophecies that shall pass because he has probably gazed into a tear from old Elizabeth's universe where she is burning down New York.

This is where things change. This Elizabeth, being influenced by her contact with Booker, recognizes the wrong in her doing but is unable to revert things. Columbia escaped her control ("as you see the lunatics run the asylum now, they don't listen to me anymore") and her bitterness and lack of hope turned her into a hell bent dictator. Imagine how bad Comstock was and Elizabeth is the same but with powers to open tears, jeez talk about a bad ass. When she repents from her wrongdoing though she remembers Booker and how he was the only one that helped her, so she opens a tear on him to bring him into the future (one of many possible ones) and show him the whole card thing and how to use Songbird to help them. This is the only way to not become her as every time Booker rescued her Songbird will in turn kill Booker and take her back for surgery.

Booker then manages to rescue Elizabeth, effectively changing the timeline there. Elizabeth then decides the only way to ensure she is not turned into that is to find Comstock and get some answers. Note that even though she is stronger and stronger as she is away from the siphon, she still doesn't fully understand how to control the tears or where do they lead.
Now here is where Comstock gets all riled up. He sees the attempts on Booker's life are not successful so he resorts to the last thing he thinks of, telling Elizabeth the truth. However Comstock believes Booker knows as much as he does (which is false, Booker is as dumb as a brick and fabricated memories to suppress the ones from Anna) and wants him to tell Elizabeth that he sold her to Comstock. In theory this is a good idea (to make him tell her) because that should alienate Elizabeth from Booker killing the small ounce of hope she has, but in the end doesn't work since Booker looses control and kills Comstock. This is when Elizabeth realizes that Booker doesn't belong in this universe (she has hints when Booker's nose keeps on bleeding as memories start to combine from universe to universe) but she stills has no idea what her ties are with Booker or Comstock. All she knows is that she is a "freak" that is not Comstock's daughter.

The turning point is given by the card from old Elizabeth, with the ability to control Songbird they can now leave Columbia forever and just rest together since nothing will hunt her. But Elizabeth has the great idea to destroy the tower with Songbird to which Booker agrees, I think mostly because that is the ultimate symbol of Columbia. In doing so the energy stored there all returns to Elizabeth and this is the major change in the story from every universe.

Now Elizabeth is effectively a god. She is able to control the tears at will, she sees all the possibilities, all the "doors" if you will and now knows exactly what happened and why. Booker does not however hold this power. Elizabeth has been taken out of the time stream, she is now omniscient and omnipresent however she decides to explain to Booker what and how it all came to this.

She takes him to the baptism because that is the turning point for the creation of Comstock, which was the root of all those troubles. When he accepts to drown himself he effectively kills 2 different timelines. The one for Elizabeth and the one for Anna being kidnapped. He is not killing his timeline though (there are infinite other universes with Booker, like one where he never killed indians, one where his wife didn't die, one where he was never presented with the option of baptism, etc) and one of those universes is the one that you get to see after the credits. As you can see on the final scene all the other Annas/Elizabeths created or affected by the baptism seize to exist as soon as Booker dies, except for the one that went with you the whole journey because she is essentially a god now and is out of the time stream, existed before and after everything but it does not mean the other Annas (timelines where Booker is never offered a baptism) need to or seize to exist.

Now as for your question of why don't the Luteces just go and kill him in the baptism. Well they do not know that is the turning point as evidenced by their dialog "well how does one know how far back to go ?". Only the now god Elizabeth knows when exactly was the turning point for it all since she sees every possibility. Same reason you don't see the Luteces inside the lighthouse with you and Elizabeth at the end.

As to only this Booker sacrificed himself. Well Elizabeth brought him to the one point where he was presented the baptism option. Like I said before there are infinite other Bookers but they are not given the baptism option so there are infinite other Annas, but not other Elizabeths, as Elizabeth can only exist if Comstock exists.