Why do natives sometimes use "would" instead of did in simple past questions? I can't find the rule about this anywhere. I hear such questions like "Why would you call the police? " or "why would he do this?" From content I know that it is not about repetitive past actions.
Would is a modal, and like all modals has a variety of meanings, generally different from the same phrase without the modal.
In this case, to me it expresses a connotation: surprise, disapproval, or disappointment: something like "I didn't expect you to call the police" or "it wasn't necessary to call the police", or even "I wish you hadn't called the police".
For me, using USA English, "Why would you do this?" is asking for my thought process. It is hypothetical; that is, it does not necessarily mean that I took the action. By contrast, "Why did you do this?" states that I did take the action, and is asking why I took that action.
If I did take the action, and you ask me, "Why would you do that?" you are asking for the reasoning behind my choice. There is no question that I took the action. We both know I took the action. Thus the question is not whether the action happened, but the reason or motivation behind the action.
Without context, the examples you provided with “would” may be interpreted in the future or the past tense.
In most cases, these phrases come along with context, so here we know you are talking about the past tense usage of “would”.
Generally, “would” and “did” can be used as past tense verbs interchangeably without any difference.
In my experience, people use both equally as much irregardless of region. There isn’t a technical difference between the two in the past tense.
I am also assuming you know that “did” is only used in the past tense.