In such fanfictions, the relationship between Naruto and Kyuubi can be developed in various ways. For example, the Kyuubi could start teaching Naruto about his heritage in a more motherly way, like sharing stories of the Uzumaki clan. Naruto, on the other hand, could show his loyalty and love towards the Kyuubi as he would to a mother. This could also impact his relationships with other characters. His friends might be initially confused but later come to understand and support this new relationship of Naruto's.
In case of an accidental sighting, the son should respect his mother's privacy by averting his eyes and exiting the area. Afterward, the mother can approach the situation calmly. She can have an open and honest conversation with the son about the importance of privacy, how accidents can happen, but also stress the need to be more cautious in the future. This incident can serve as a learning opportunity for both of them to strengthen their understanding of personal boundaries within the family.
A frequent trope is the emotional bond between them. Naruto might seek comfort from the Kyuubi during tough times, and the Kyuubi would be there for him. There could also be a trope where Naruto tries to make the Kyuubi proud, similar to a child trying to impress their mother. He would constantly strive to become stronger and better at using the Kyuubi's power in a positive way.
He should respect his mother's privacy and act as if he didn't see it. It would be very embarrassing for both of them if he made a big deal out of it. If possible, he could try to avoid situations that might remind him of it in the future. And the mother should also be more careful with her private things in the future to prevent such accidental exposures.
One key moment could be when Naruto gets seriously injured and Tsunade heals him with great care, showing her motherly love. Another might be when Tsunade gives Naruto some important advice about handling political situations in the village, like a mother guiding her son. And also, when Naruto defends Tsunade against someone who disrespects her, showing his loyalty as a son would to his mother.
Yes, if it's a rooster. It could symbolize new beginnings. In many cultures, the rooster's crow is associated with the start of a new day, so in the mother's story, it could represent a new phase in her life or in the family.