My mind is blown.
I am so confused... it's all so crazy. I don't get. What happened? What is the connection with Rapture? So Elizabeth ceases to exist in the new universes and therefore every universe? Is there a Columbia? What happens with those damned twins? The only time I have felt like this is when I watched The Prestige.
I am really sad the game ended. I was masterful. Among the greatest games I have ever played.
1) There's really no connection with Rapture (only theories like NotStan mentioned), it's there to show that there are multiple universes, Elizabeth said: "There's always a city and a man" mentioning that Rapture and Andrew Ryans existed in multiple universes, the same people, yet different.
2) Elizabeth and Comstock ceased to exist from every dimension. Comstock "born" on the baptism, so if Booker died there, Comstock could have never been born. At the same time Booker also died, so Anna/Elizabeth ceased to exist because his dad doesn't exist now.
3) Yes, Columbia did existed. But at end since Comstock died, Columbia (along with Comstock and Elizabeth) ceased to exist.
4) The "twins" aren't really twins. Rosalind Lutece was hired by Comstock for the interdimensional travels, during her research she found a particle that let her speak with her other half in another universe (which is Robert Lutece). They both start experimenting until both found a way to travel to their respective universe, that's how Comstock was able to take Booker's daughter.
But to answer your question specifically, the twins were murdered by Comstock, but they were alive in other universes and they wanted to take revenge for Comstock's murder, I also think they were looking for redemption for what they did to Booker. If you are asking that what happen to them after the ending, well they probably still living in multiple universes without problem.