No. In the official Naruto series, Naruto is the son of the Fourth Hokage, Minato Namikaze. Fanfiction might play around with this concept for creative or alternate universe purposes, but it goes against the original lore.
In this fanfiction, Naruto might save the Third Hokage by using his newfound powers. Maybe he taps into a deeper level of his chakra and arrives just in time to fend off the enemy attacking the Third. His determination and growth as a ninja would play a crucial role in this rescue.
The Third Hokage's training might have introduced Naruto to more advanced jutsus. For example, he could have taught Naruto better control over his chakra, which is crucial for performing ninjutsu. This would allow Naruto to use his Rasengan more effectively and with less strain.
One challenge could be mastering difficult jutsus. The Third Hokage might set high standards for Naruto to learn complex techniques, which would be a struggle for him.
Well, in certain fanfiction stories, Naruto may hate the Third Hokage due to the way the Hokage handled the Uchiha situation. Since Naruto has friends like Sasuke who are Uchiha, he might resent the Third Hokage for not preventing the downfall of the Uchiha clan, which in turn affected Sasuke deeply. And he may see this as a sign of the Third Hokage's incompetence or lack of real care for the village's well - being as a whole.
Yes. In some fanfictions, Naruto could be depicted as having great admiration for the Third Hokage. The Third Hokage was a highly respected leader in the village. Naruto, growing up in the village, would have heard many stories about his leadership, wisdom, and sacrifice. This could lead to him being a fan in the context of fanfiction.
If Naruto was the Third Hokage's son in fanfiction, his social standing in the village might be different. He could have had more privileges and perhaps less of the ostracism he faced in the original story. The Third Hokage might have been more directly involved in Naruto's upbringing, which could have led to Naruto being more trained in traditional village techniques earlier.
In this fanfiction, Naruto could be given a special mission by an unknown entity to save the Third Hokage. This mission might push Naruto to his limits and force him to unlock hidden potential within himself. Along the way, he might face challenges that test his loyalty and friendship.
Naruto would probably be extremely hurt. He has always looked up to the Third Hokage as a grandfather figure. So, initially, he would be in denial, thinking there must be some misunderstanding.
Perhaps it was related to Naruto's heritage. The Third Hokage might have been influenced by some hidden secrets about Naruto's parents. If there were things he didn't want to come to light, he could have betrayed Naruto to keep those secrets buried. For example, if Naruto's father had enemies within the village, the Third Hokage might have tried to keep Naruto at a distance to avoid those old grudges resurfacing.