Why would you know this? Are you a historian or something? Are you secretly Dennis Miller who knows every piece of historical minutia known to man?
Alice Roosevelt was a pretty significant figure; she was one of the first American "socialites" to speak of. She got in trouble for driving in a car!!! and speeding (going 20 mph!!!!). She got around a bit, was always seen at openings, and eventually married a rising star congressman. She was a constant source of stress for Teddy. That portrait in particular is pretty famous.
Why did I end up in rapture?
She drowns Songbird by teleporting him there, but other than that there's no particular significance. Elizabeth explicitly tells you why you're there.. In every dimension, in every universe, there is a man with a vision who builds a city. The lighthouses are the nexus between dimensions. She brings you to Rapture and surfaces you in a Bathysphere so you can see the lighthouse in that dimension (the BioShock 1/2 dimension), and then when you go through the door you see all the lighthouses--all the dimensions.
Why did they kill me at the end?
In some dimensions Comstock is Booker. When he was baptized after Wounded Knee, he took a new name. In order to prevent everything that happened and save Elizabeth's suffering, the only way to do it would be to kill Comstock as he was being born--so they go to the moment where Comstock was born (Booker's baptism) and drown him, so that he is never born.