There could be. It depends on the specific comic universes and story arcs. Sometimes, crossover events or special storylines might include this kind of training scenario.
Yes, Batman comics have shown Bruce Wayne as a boy on several occasions. These depictions often explore his early traumas and the events that shaped him into the hero we know.
Jason could show care for Batman in fanfiction by having deep, heartfelt conversations with him, revealing a hidden soft side. He might also take on dangerous tasks alone to spare Batman from harm or work behind the scenes to gather crucial information that benefits Batman's endeavors.
Not exactly. Batman Begins is one interpretation of Batman's origin story, but it's not the definitive or only 'real' one. Different Batman stories and adaptations offer various takes.
Well, Batman's origin story often involves the trauma of witnessing his parents' murder. This event drove him to train and develop the skills and gadgets needed to become the Dark Knight.