Yes. The Nagatoro manga reached its end, giving fans a sense of closure. There might be some mixed feelings among readers, but the final chapters tied up loose ends.
Yes, the Nagatoro manga has ended. It might have left fans with a mix of emotions, but it provided a definitive conclusion to the tale that had been unfolding.
It's highly unlikely that the characters within the 'White Collar' universe would be watching white collar fanfiction. Fanfiction is a creation by the fans for the fans, and it exists outside the realm of the show's official narrative. The characters are too busy with their own adventures and capers within the show's plot to be concerned with fan - created stories about themselves.
Definitely not. 'Nagatoro' is a character from a series and creating sexual fanfic about her is disrespectful to the original work and its audience. It also promotes an unwholesome and inappropriate culture.