Sorry to hear of your situation. I've had trouble with the one ABOVE, but one of my friends had some problem with their neighbor BELOW with their TV noise and all that...
Just some thoughts:
- Kids run around, that cannot be helped. But as they live in a building shared with others, they would be obliged to "do their best" to keep down the noise, including disciplining the kids properly, and/or putting carpet to keep the vibration and noise down. It could be that they are not at all aware of the noise they are causing. And those wood (or wooden-looking) "flooring" transmits noise like nothing else.
- The first step in these circumstances is just to go and talk peacefully, assuming that they are not aware of the noise they are causing. For this if you just go and talk friendly, that might be OK, and that might solve the issue.
- If they take the "so what" attitude, then it might be an idea to talk to the landlord or go together with the one below them. "Some" people think it's their right to do whatever they want to do in their unit - too bad if that generates some noise but so what - so that it might be more effective if you could talk together with another neighbor. Or if there is an administrator or the landlord you can rely on to help talk to them, it might be helpful.
- About the late hours - I assume it is a rented property? Is there any way you can check what kind of general rules they have? (like be careful not to make noise after 10pm, avoid taking a bath after 10pm, that kind of thing?)
1. すみませんが、ちょっと おねがいしたいことが あるのですが。
(Excuse me, but there is something I'd like to ask you.)
むすこさんが、へやのなかを げんきに はしりまわると、わたしの へやまで かなり ひびく の です。
(When your son runs about the apartment, it resonates quite a lot to my room.)
うえに ひびくのは いがい かも しれませんが、おとと しんどうが かなり します。
(You might find is unusual that it vibrates above, but there is quite some noise and vibration.)
2. まえに こちらに おすまいだった ごかぞくも、ちいさな おこさんが ふたり いました。
(The family who used to live here had two small kids too.)
でも、おとが きになった ことは ありません でした。
(But it never happened that their noise bothered me.)
ですので、もしかしたら、リフォームの せいで、おとが まえより ひびくの かも しれません。
(So it might be that the reform has made the noise carry more.)
ただ、おねがいしたい の ですが、かなり ひびきますので、きをつけて いただけませんか。
(But in any case, I really would like to ask you to be careful, as the noise is quite a bit.)
3. きょうどうじゅうたくに すんでいる の ですから、わたしも もちろん、できるかぎり きを つけて ごめいわくに ならないように します。
(As we live in shared housing, of course I will do my best not to inconvenience you.)
うえからの おとで もし きに なる ことが あったら、えんりょなく おっしゃって ください。
(If any noise from above you bothers you, pleas don't hesitate to tell me.)