Byakuya remains alive throughout the manga. The author chose to keep him in the story for various narrative purposes and to continue his character development.
Byakuya doesn't die. The manga plotlines and developments always find a way for him to persevere and stay in the storyline. His skills and determination help him survive the tough circumstances presented.
There is a good chance that such fanfiction exists. Given the popularity of the Bleach series, fans love to delve into different character interactions. Byakuya and Ichigo have their own unique personalities and storylines, which makes for an interesting premise for fanfiction writers to create stories about Byakuya having feelings for Ichigo.
He might be cold and distant towards Ichigo. Byakuya could give him the cold shoulder and be less cooperative in situations where they have to work together. For example, in a battle situation, he may not offer Ichigo as much support as he should.
Byakuya's personality is one of dignity and reserve. This affects his love for Rukia in that he doesn't easily let his feelings show. He is a man of few words, and when it comes to his feelings for Rukia, he is no different. But his love is manifested in his protectiveness. His noble upbringing means he has a code of honor, and part of that honor is now to protect Rukia. His love is a quiet, unassuming one that grows as he starts to understand Rukia more and more as a person, not just as a member of his squad.
It might mean that Byakuya trusts Yachiru with something very powerful. A monster is not an ordinary gift, and it shows that he has confidence in her abilities to handle it.
In fanfiction, if Naruto is the reincarnation of Byakuya, the plot takes a huge turn. Firstly, Naruto's character development changes. He may inherit Byakuya's sense of honor and pride. Secondly, the power system gets a major overhaul. He has access to Byakuya's Zanpakuto and its abilities. This would also attract different enemies. Instead of the Akatsuki, he might face threats from the Hollows or other Bleach villains. Moreover, his training would be more about spiritual growth rather than just physical and ninjutsu training, which would completely reshape the Naruto plot as we know it.