I still don't know what Song Bird was. I know he was Elizabeth's protector. But is it ever discussed what the origins of it was? Just who was it. I'm assuming it was an automaton of some kind, but was it a robot or a person inside of it?

Songbird was created by Jeremiah Fink to be "both less and more than human." That's about all I could figure out from the audio logs I found. There's a small boatload I missed. I think I was about 20 short of the achievement, so there might be more.

So the game is stuck in a sort of infinite loop. Booker Dewitt took part in the battle of wounded knee and killed a lot of native americans and basically saw some shit. After the battle he felt really bad for what he did so he went to go get baptised of his sins but just before getting baptised he refuses. He then slips into a life of alcoholism and gambling and at some point has Anna. It's not really explained how and no mother is ever revealed. Then one day the Male version of Lutece comes and asks for the baby offering to wipe Booker's slate clean in doing so. Booker hesitates but offers Anna to the man to wipe away his debt. Booker having felt really bad for what he did tries to get Anna back but is too late as Comstock and the Luteces pull Anna through a tear which ends up severing her pinky. Booker gets really depressed at this point. He carves AD on the back of his wrist which means Anna Dewitt. Then at some in the future he returns to the preacher to be baptised of his sins and then changes his name to Zachary Hale Comstock.

This is where things get a little confusing to me. I think Booker now Comstock gets in touch with Rosalind Lutece and her twin brother finding out that they are the same person but from different dimensions. They demonstrate that they can create tears into alternate times and show Comstock how this is done. He gains foreknowledge of the future because of this, and is able to become a prophet of sorts because he can predict the future through these tears. Somehow Comstock is able to gain a fortune of money and decides to build Colombia. This is where I'm not sure of things. I don't know why he chose to build Colombia and I'm not sure where is jingoist philosophys come from and how he manages to create a society that basically worships the founding fathers as Gods since it's never really shown much in his past.  I'm not sure if I missed this part. But basically he is seen as a prophet by the people in this new paradise because he can see into the future. He at some point gets married to Lady Comstock as well.

So the Luteces and Comstock do some research for a while and find an alternate Booker from the past that has Anna. This is when Comstock hatches his scheme to take Anna away from that Booker and to bring her back to him. So he sends Robert Lutece back in time into that alternate world to wipe away that Bookers debt and basically take Anna back for his own. I'm not sure whey he renames her Elizabeth, maybe to through Booker off into thinking she's not Anna? He then hides Elizabeth away in the Angel statue and makes Song Bird protect her because he knows that Booker is going to come looking for her thanks to the helps of the Luteces. The Luteces then experiment on her which awakens her powers to create tears. They create the Siphon to limit her powers so she can't just up and leave Colombia. Comstock then he creates the false shepard story to turn his people against him and protect Elizabeth.
Lady Comstock now suspicious of Comstock starts questioning where the baby came from and even blames Rosalind for an affair. Rosalind explains where the baby came from but Lady Comstock doesn't believe her. At his point I'm not sure if its Comstock or his wife that kill the Luteces. But I know that Comstock kills his wife and blames it on the Vox who are starting to uprise around his city and going against his nationalistic racial pride. Comstock is also aged considerably due to messing with all the tears so he now appears as an old man when he should only be 38.

The Luteces some how get stuck in a world between life and death and can pretty much appear anywhere at anytime perhaps due to there mastery of figuring out how to use the tears. Feeling sympathy for Booker they go back in time to just before he was baptised and bring him to 1912 in order to save Elizabeth from himself. Now as displayed there could be hundreds and thousands of versions of how Booker's adventure through Colombia goes. One is where he dies while joining the Vox and never makes it to Elizabeth. Another is where he has to get tools for Cheng and another is where he has to get weapons for the Vox. Another version shows Booker never returning to save Elizabeth and she therefore orders an attack on New York City due to his prophecy that he had. The infinite dimensions create tears that happen all over Colombia. Anyway Booker goes on his adventure in your playthrough and finds out the truth, and the only way to stop this infinite time loop is to kill Comstock, and he achieves this by having Elizabeth drown him before he can become baptised as Comstock.

After the credits I think it is implied that Booker is in a world where he has Anna.

One thing I'm wondering is does the old Comstock have the A and D carved into the back of his hand as well?