Have I done sthg wrong? or Did I do sthg wrong?

1. Mainly British
2. Mainly American.

You can use "something" or "anything".

You can use either. It is not an issue of BrE versus AmE, although it can be an issue of greater formality or a lessened formality which leads to the notion it is a BrE versus AmE thing.

There are a number of uses for 'have/has' + PP in English.

You are familiar with the PP of Experience - Have you ever snorkeled?/Have you [ever] been to Venice?

the PP of Continuation - How long have you lived in that house?

One of the harder uses for EFLs to grasp is the use of 'has/have + PP to describe a Hot Topic/Importance.

All these [there's one more that escapes me at present] uses have an important similarity, they describe an "up to now/up to the present time".

That is where your example lies and you can use either, the choice is highly personal and there are many variables that influence the choice, most of a personal nature.

1) I haven't heard from you for/in a while. Have I done something wrong?

Illustrates more formal, illustrates a greater importance in the speaker's mind, ... .

or

2) I haven't heard from you for/in a while. Did I do something wrong?

Illustrates less formal, illustrates the speaker is saying it's of lesser importance.

The reason that this is so complicated is that it is based solely on an individual's own personal feelings. The issue could be of utmost importance but the speaker might downplay it simply because they are embarrassed.

I am sure I have posted this in one of Paok's threads before, but here goes...

Speakers of American English generally use the present perfect tense (have/has + past participle) far less than speakers of British English. In spoken American English, it is very common to use the simple past tense as an alternative in situations where the present perfect would usually have been used in British English. The two situations where this is especially likely are:

1. In sentences which talk about an action in the past that has an effect in the present:

American English (AmE) / British English (BrE)

Jenny feels ill. She ate too much. (AmE)
Jenny feels ill. She's eaten too much. (BrE)
I can't find my keys. Did you see them anywhere? (AmE)
I can't find my keys. Have you seen them anywhere? (BrE)

2. In sentences which contain the words already, just or yet:

American English / British English

A: Are they going to the show tonight?
B: No. They already saw it. (AmE)
A: Are they going to the show tonight?
B: No. They've already seen it. (BrE)
A: Is Samantha here?
B: No, she just left. (AmE)
A: Is Samantha here?
B: No, she's just left. (BrE)
A: Can I borrow your book?
B: No, I didn't read it yet. (AmE)
A: Can I borrow your book?
B: No, I haven't read it yet. (BrE)

That certainly describes a reality, which I acknowledged, but it doesn't describe the reasons.