1) You seem to use "classmate" as a blanket term for any ninjas around the same age range. Which is fine, as they likely would have been classmates. Except, within a class year, they're further broken down into teams. Specifically, Naruto and Sakura aren't just classmates, they're teammates. Consequently, teammates typically have stronger bonds with each other than they would with other classmates. Tsunade and Jiraiya were also teammates, which is part of why he'd be in a position to rub off on her: they know each other very well. So it feels off to hear Naruto referred to as Sakura's classmate, and equated with other classmates. That would only fit if this was very early on, before or right after they were put on the same team. And you can tell by Naruto's clothing that this is long after they've been teammates for some time.
2) You wouldn't say "Tsunade Hokage," Hokage are referred to by their names or titles. So your options are: Lady Tsunade (Tsunade-sama) or Lady Hokage (Hokage-sama). Sometimes Hokages are called by their number, such as Lord Sixth or Lord Second, which would make her Lady Fifth. Also, Naruto is very informal in general, and would never called Tsunade "Sensei", even if she were teaching him directly. He calls her Granny Tsunade, which started off as something of an insult, but it's more a term of affection now.