The relationship can start with respect. Since Zangetsu is a powerful entity, Ichigo would initially respect her strength. As they interact more, like during training sessions, they can start to share personal stories. This sharing can lead to a deeper emotional connection.
One possible plot could be that Ichigo, after fusing with Zangetsu, gains new and enhanced powers. He might have to face a powerful new enemy that only his fused form can defeat. Maybe he struggles at first to control the combined power, causing some chaos in the Soul Society.
Perhaps in the fanfiction, Female Zangetsu has been separated from Ichigo due to some dark magic. Ichigo then has to embark on a journey to find her. Along the way, he discovers more about her past and her true nature, which deepens his love for her when they are finally reunited.
Rukia would probably be both amazed and a bit worried. Amazed at the new power Ichigo has, but worried about how it might change him. Chad would see it as a natural progression of Ichigo's power and be supportive. Orihime might be a bit scared at first, but then she would see the good in Ichigo's new form and try to encourage him.
Yes, there are likely fanfictions out there with this concept. The Naruto fandom is vast and creative, and crossovers or unique concepts like Naruto having White Zangetsu are not unheard of. You can start by searching on popular fanfiction websites.
One plotline could be that Naruto finds Zangetsu during a mission in a hidden part of the ninja world. He then has to keep it a secret from his village while he tries to figure out how to use it. Another could be that an enemy uses some sort of dark magic to force Zangetsu onto Naruto, and he has to learn to control it to break free from the enemy's control.
Perhaps in this fanfiction, Naruto is given White Zangetsu as a gift from a mysterious figure. With it, he discovers new powers and abilities. He might have to train hard to master it, and in the process, he could face challenges not only from his own world but also from forces that are attracted to the power of White Zangetsu. This could lead to new alliances and enemies for Naruto, completely changing the dynamics of the Naruto universe.
A common theme is the exploration of their dual natures. Since Ichigo has his own powers and the female hollow ichigo has her unique hollow characteristics, fanfictions often play with how these two different yet connected aspects interact.
I'm not sure of the exact name, but there's a fanfiction where female hollow ichigo gets separated from Ichigo during a battle. It then follows her own adventure to find her way back to him, while Ichigo is also searching for her. It's a story full of action and their unspoken bond.
It could be really interesting. Naruto is known for his strong will and determination. If he had Zangetsu, he would likely try to master it quickly. Since Zangetsu has a mind of its own in the Bleach series, there could be some fun interactions between Naruto and the sword spirit. He might even use Zangetsu to cut through some of the toughest jutsu barriers or opponents that he couldn't defeat with his normal techniques.