Who is right here?

I prefer to play PS4 with my grandma because I'm never disturbed there. (She lives in a skyscraper).

A neighbor who lives one floor below has "optimal hearing" and is therefore mentally ill. She feels annoyed by the roar of my Playstation and the sound effects of my headphones (however she can hear them?).

Now this woman (neighbor) has turned to the high-rise administration and I was snapped at by a guy who forbade me to play at my grandmother's Playstation.

He said he'll call the police if I don't take my Playstation home with me.

I pay attention to the quiet times in the house and no one (not even her husband) except she can hear me.

The question now is whether I'm really not allowed to play with my grandmother because of a mentally ill woman who should be in the slap anyway?

If you have been shown to use headphones, every step from the outside will be difficult. The question is whether you want to piss off your grandmother's neighborhood because of a simple video game console.

It's about people wanting me out of here. I think the woman simulates, because there are days when she complains even though I haven't played…

Yes, you can expect the volume to be reduced. You can no longer talk about visiting your grandmother if you just spend hours playing with your grandmother.

Either you should be right and hey get a pc they are quieter, i'm happy to put one together for you and if they hear the pc i swear i'll come over and they'll get broken ears

Of course that's a visit.

The neighbor has no chance.

If you keep the rest times, that's okay for now. You can generally be with your grandmother as often as you want - only if you would live permanently there (or live temporarily, so also stay overnight and so) that could interest the landlord with regard to ancillary costs - but what he can't say about it.

What you do when you visit your grandma is only your and your grandma's business, as long as you don't make any noise. And even that - in certain ramen - must be tolerated at normal times. Playing children can't simply be forbidden.

The police don't care about that during the day either, or they will come, find that it is not loud, and leave again.

Is it a visitor? There are no rules as to how often or how long you visit. As long as you don't live there permanently / for days (sleep, shower, etc.) it is a visitor.
Even if it is not a visit, this is only an argument with regard to additional costs (heating / water) where the landlord could possibly demand higher shares. He can't forbid it. You can live with whoever you want.
Volume is acceptable during the day in the usual ramen. Playstation & Co are one of them. If he takes headphones even more.

So she can ask friendly questions, but she can expect that just as the sun shines on Sundays.

When he puts on headphones he can't be heard either. But if he screams loudly (that's how my son does it), then the neighbor doesn't have to accept it.

A certain noise level is always reasonable during the day. How loud exactly the questioner is or what can be heard despite headphones, but nobody can say from a distance. According to the question, apart from the one neighbor, nobody ever heard anything or complained.