Yeah I kinda agree with that.

I don't remember a single moment when Booker says he is innocent. He was a mercenary and that is explained when you go to the hall of heroes., he killed a lot people before the events of infinite.

did you like the game?i love it.mainly the story.finished it ages ago.

If anyone reads this and I'm wrong, please correct me, a beat this almost a year ago, so it's not all that fresh on my brain. Yes, Booker is Comstock, kind of. They're two different versions of the same person. Comstock and Booker diverged after the battle of wounder knee (may be the wrong battle), and Booker chooses to get baptised. In one reality, Booker runs from the baptism and remains Booker. In the other, accepts the baptism and becomes Comstock. Anna and Elizabeth are actually the same person, not divergent people. 

Now I'm going to the Luteces, because nothing else will make sense if I don't explain this. So the luteces aren't actually brother and sister. They're the exact same person, and their only difference is one has an x chromosome and the other has a y chromosome. They were born in different dimensions. They both became physicists. The girl Lutece discover how to jump between dimensions a few years before her "brother" would have. She discovered her "brother" in another dimension and contacted him. Eventually they met Comstock. Comstock used their technology to creat Columbia. All of Comstock's work around these device made him sterile. He knew of his "daughter" (Anna renamed Elizabeth) in Booker's dimension and had the Lutece's buy Anna from Booker. Booker went back on his decision, which is why he stole Anna instead. The Luteces regretted their decision, so they are trying to help you get Anna back. An interesting thing that you probably didn't notice in the game. When they ask you to pick heads or tales, and the brother has the chalkboard on him. All of the markings represent every time Booker has gotten that far. So it is thought that the Luteces have done this 90+ times with 90+ Bookers who have died. This aldo adds another interesting idea, where each time you die in the game could almost be thought of as one of the times Booker fails. 

Now to answer your last question. The Luteces appear everywhere because they jumped between so many dimensions, and they became seperated between them. Remember all of the people that were in a state of being alive and dead? Same idea. Basically, you're seeing a shadow of what they once were throughout the game.