'Have you received it?' or 'Did you receive it?'

Grammatically either. Tonally - maybe the former is softer and slightly more formal or neutral. The latter is more direct and depending on the context might be considered a little firmer. But that's very subtle and highly subjective. eg, I'd use 'have' for a friendly enquiry as to whether it has hit someone's inbox or not. I'd use 'did' if I was checking why they hadn't responded or taken action.

Personally I'd say "Did you get it yet?" or "have you gotten it?"

Either is fine, but fairly formal. In most casual settings I would say “Did you get it?”

Also consider that these phrases can be interpreted in the following ways: Have you received it? (It might still arrive). Did you receive it? (The time for receiving it has finished).

Prescriptively speaking, I think the present perfect is more appropriate, as it connects the past action (sending the email) with a present result (email received or not). Descriptively speaking, many American English speakers use the simple past where other dialects use the present perfect.

Either is fine.

There is a very very very slight nuance. "Have you received it?" means they have been waiting for a something and want to know if it has arrived because it may or may not have gotten there yet. (like an email with a large attachment) "Did you receive it?" means you expect that the message *should have* arrived already want to know if it ever actually got there.

Sometimes present perfect and past simple are interchangeable, like in your case. But if you want to emphasize the result of the action, you use present perfect.

I think both are acceptable

The first is in present perfect tense and the second is in simple past tense. Based upon your preference, choose either.

If they sent it just now or a few minutes ago - have