It's a mansion by the way and they just moved in below. Yes, this is from below (unusual), not above. The previous family, with more young kids than this one where silent as a lamb.

Cultural: Is it acceptable for the family to allow their 5-yr old son to run around the mansion, pounding on wood floors as he does so?
Last night at midnight, they returned home and the son was going crazy with this. At midnight?? Can the parents be that obtuse?

Language: Could you help me express some ideas as below? Could you give answers in kanji with the hiragana either on top or additional version?

1. [calling at the door] I'm sorry to disturb you but your young boy is running around inside the room/apartment and it's causing an unbearable noise in my room. Really, the pounding and vibration are quite terrible.

2. It's strange because I never heard noise from the family who lived there before and they had 2 very small children. So, perhaps because of the reform work the noise is carrying more. But could you please do your best, as it is really disturbing me.

3. And by the way, I try to be very quiet and considerate, but if I make too much noise, please don't hesitate to tell me about it.

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.)

You can refer to the possibility that the wooden floor is making the situation worse, without such a comparison.
And, please don't forget that even in a shared housing, a resident has a right to make alterations to his/her room to some extent.
I mean, what might have a problem is the wooden floor, not the alteration they made.

their 5-yr old son to run around the mansion, pounding on wood floors

It's nothing but my assumption, but he might happen to have a certain kind of difficulty in controlling emotion.
(I presume a parent of a child with such a difficulty may well wonder when to confess it to new neighbors.)
Of course it's a nature of little kids to run around, as is mentioned by AK.

My family is one of the oldest in the neighborhood.
Being residents in a shared housing, we experienced some neighbors' moving in / out, but fortunately we neither did give nor were given a complaint related to noise.
What I think as important is that you don't make the family misunderstand that you are trying to exclude them, taking this "opportunity" of the noise problem.
I recommend keeping the balance between "we are basically welcoming you as new neighbors" and "we are really annoyed by the noise."

In Japan, those who move in are expected to come and greet their new neighbors. If it's just been days since they moved in, they might soon be coming to your door to say hello, and that would be a great oppurtunity to discuss about the noise. Or you can seek for a chance to "bump into" them in the hall, so that it would be natural to talk about your problems.

They also might need time adjusting. People tend to get restless and make noise when they've just moved in, but soon they settle down.

The other thing is that you can also try talking to the other neighbors first. If it's that kind of noise, I'm sure the other neighbors are having problems with it too. Some say that, in order to avoid complications, you should go straight to the kanri-kumiai, in other words the board, and let them distribute a note without mentioning the name of the claimer. I used to be on the board and indeed a lot of residents requested me to do so, in which I did. It's up to you.

AK: Thank you very much for your very helpful translation and advice. Your efforts are appreciated.

Omotenashi: I tend to agree with you about leaving out the previous family. I felt uneasy about that part anyway.

But...

Let me clarify, if I already did not, how bad this situation is now of today:

-The noise is getting worse and almost constant all day through to midnight.
-The family returned home midnight one night and started up the noise (running, jumping kiddy) at 11pm another night.
-This boy is 5 years old!? At these hours? Ok, parents call, but NOT when such poor behavior is a nightmare of inconsideration to neighbors.
-I have already had to change the room I sleep in. But there ISN'T a room where the unbearable noise doesn't explode into, so there is NO escape.
-I have not slept properly for 3 nights. I have to get up at 4am tomorrow and now, at 8.30 the toddler is pounding and crashing around. I know this won't stop before midnight. He also seems to be now playing ball games inside the apartment as I can clearly hear a bouncing ball. For goodness sake!
-How do you imagine my stress levels are as a result of this? My friends already say I'm looking quiet, dark and withdrawn. And I mention these points because you appear to lean towards 'live and let live' when this family are not letting me live as I did a week or so before the sad day of their arrival.
-There is NO way to mask the noise. TV, headphones, music - all just emphasize the bass crashes and thumping from below.
-Finally, I think that to move into a new place and immediately invade the (small) block in this way is just plain arrogant and selfish, and I'm not optimistic about resolving this to give me some of my space back.

Uco: I'm going to speak to them soon, maybe tomorrow, and then contact the block management company.

I'll let you know what happens over the next few days.

Thanks again all.

I'm sorry to hear of your situation.

Just to update those who very kindly gave help and advice to me with this problem.

I spoke to the neighbors and also contacted the block management company.

This was about a week ago. From the next day, it was as if the offending apartment was vacant. Not a sound for over a week. I could sleep again, after 3 days of hell, and I now have my previous space back to enjoy some quiet time during the day and, when working, after a hard day.

Why on earth would I feel I have to move out when it is not me being the anti-social neighbor?

First of all I am glad to hear that your problem was solved and now you can sleep without disturbance.

I hope you will not have another noisy neighbor moving in any more.

Many people probably have a similar problem as long as they live in an apartment, shared housing. I usually talk to landlord when a similar situation comes up but that haven’t really worked very well. It takes a certain amount of time for noisy people to change their life style or habit. Then I usually have to put up with the disturbance. Then if putting up with it becomes totally unbearable I would move out even though I know I might encounter noisy neighbor at the next new place.

I thought that talking directly to the new neighbor that they are bothering you would not be a good idea because that would make your relationship with them awkward. I feel that I could complain of the noise to people as long as I am really familiar with them or a close friend with them. Or like others pointed out you can talk to your other neighbors first as they might have a same problems as yours toward noisy neighbor. However it is hard for people to control other people’s behavior and if you really cannot tolerate, you might want to move out.

In my previous apartment, my neighbor right side next door had a dog that burked so loudly all day. The residents were not supported to have a pet but nobody warned them because many other neighbors secretly had a pet too. My left side next neighbor was a couple that always argued and having a fight all day with dirty words but especially between 2-5 AM in the morning (when people are sleeping), their fight would reach a climax. The wife would loudly shout and calling for a help (help me~ help me~ I am killed~~!!), it sounded like her husband were trying to strangle her, throwing something at her and kicking and punching her. One day eventually some other neighbor in the apartment called a police early in the morning (about 4 AM when they are having a fight) and when police arrived they created one more scene again with the police but the husband was eventually taken to the police station. My downstairs neighbor was also a big problem. They hold laud party almost every night after might night and once I called a police but the situation was not overall improved. One day at about 3 AM in the morning they started s◎x at their back balcony outside, just right above my bed room. I put on my earplugs deeply and I gave up. I had had enough and moved out.
Now I live in an apartment that is used to be an office complex. They change it to residential apartment. But actually I still have some noisy neighbors here:). My neighbor upstairs seems they sometimes hold an aerobics exercise session or something, I hear loud footsteps and people’s voice all day long. My neighbor next door listens to the music loudly and its base notes always bother me. But I still think that moving to this new apartment has been a better choice than staying in a previous apartment. And, earplugs are always my best friend.

I hope you understand that sometimes moving out can be a good last resort. Anyway your problem has been solved and that is really nice!