Sometimes I see the sentence
Have you done something, yet?
Is it correct to write it that way? If not, what would be correct? If it is correct, why is it?
Yes. The usage of yet in your sentence is correct, both grammatically and in its meaning, but that comma should not be in there.
"Yet" is an adverb which is synonymous to "already":
Have you done this already?
Not entirely sure what you mean by something.
If something stands for some verb, as in "Have you verbed yet?", then it is correct (its position in a sentence).
If you're asking about this very sentence "Have you done something yet?", then I'd say it'd better to say"Have you done something already?". The words "something" and "already" both imply "yes", whereas "anything" and "yet" don't - the answer could be yes or no.