Some of these are just cultural. It's the sort of thing most Japanese youth of that age are familiar with, which is often the target audience. That's outside of anime as well.
The school and club settings in particular. There's a larger focus on school life in Japan for that age group than in the west. They spend more time there and it's a larger part of their lives, so it gets more focus in shows that revolve around that age group.
The '2D girls' guy gets used a lot recently, but it's a commentary on a real cultural phenomenon, what some would even consider a serious problem for some youth in fact.
The sex jokes and body humor too, it gets old, but that's what passes for general humor in Japanese culture, that and puns. Metric tons of fucking puns. They're also not alone in liking raunchy humor, they're just more open about it and it's not considered as 'adult' as it is elsewhere in the world. Playing off of insecurities for humor is also not unusual in general, and isn't limited to Japan. They do often reuse the same jokes over and over is an issue, but I've also seen some pretty damn funny variations that go outside the box. The example gif they used was pretty damn funny in context for example.
Only youth can save us is also not really a cliche so much as a social teaching tool. It's moralistic and not limited to anime anyway. Lots of shows with young protagonists, especially young 'warriors' do that all over the world. It's a storytelling tool meant to provide a platform to teach kids about believing in themselves, self reliance, and other moralistic lessons. That's just youth entertainment in general.
What bugs me is when Anime goes out of it's way to point out that it's an old cliched joke, and then does it anyway throughout the show as if pointing it out somehow makes it subversive.