Maybe Carlisle wipes Bella's tears because he sees her as a vulnerable human in a dangerous supernatural world. He has a kind heart and wants to comfort her.
Often, it starts with an unexpected encounter or a moment of realization. Maybe Carlisle sees Bella in a new light, not just as Edward's girlfriend but as an individual with her own charm. Then, there are small gestures of kindness and understanding between them that gradually build up the relationship.
One possibility in such a fanfiction could be that their relationship completely alters the dynamics of the original Twilight story. For example, it might change how the Cullen family functions and their interactions with other characters. Maybe Bella's connection with Carlisle would lead to a different approach in dealing with the Volturi.
The Cullen family members are important characters too. Edward, for example, might have an interesting dynamic with Bella in this fanfiction. He could be either protective or jealous of Carlisle's special relationship with her.
At first, it would be a bit awkward as Bella adjusts to her new life. But Carlisle would be very patient and understanding, like a father figure. Esme would be more of a motherly presence, always there to comfort Bella. As time goes on, they would become closer. Carlisle might teach Bella about the vampire history and ethics, and Esme would share her love for the family and home. Eventually, they would have a very strong, loving bond.
There could be Edward. Even though the focus is on Carlisle and Bella, Edward's presence might still be felt as he was an important part of Bella's life before. His reactions to Carlisle and Bella's relationship could add some drama.
In the fanfiction, Carlisle's sense of duty to his family and the vampire code might be a hurdle. He has to balance his feelings for Bella with his responsibilities to the Cullen family. Also, Bella's own human life and her relationships with other humans, like her father, could be a complication. How can she maintain those relationships while being in love with Carlisle? There's also the issue of secrecy. Their relationship has to be kept hidden from the outside world to protect both of them.
I would doubt it. Given the rather odd nature of the idea of Carlisle changing Bella's diaper, it doesn't seem like the kind of fanfiction that would appeal to the general 'Twilight' fan base. Most fans are more interested in the romantic, vampire - related aspects of the original story, not such a bizarre situation.
In this fanfiction concept, it would completely change the dynamics of the Twilight story. Bella being Esme and Carlisle's daughter would mean a different kind of family relationship within the Cullen family. For example, her relationship with Edward would be that of siblings rather than lovers. It could also lead to different story arcs regarding her powers and how she fits into the vampire world. There could be new conflicts and alliances based on this new family connection.