The first implies that he quite literally crosses the river by boat or some other specific means. The emphasis of the second sentence is on the school. It's across the river and he can get there any way. He can even fly there. It's a subtle difference.

No. You could say though The boy gets to school every day by crossing the river. The boy goes across the river every day to get to school. But, of course, using the verb ‘to cross’ is best.