What did you do there with him? should be accepted. I have read all of the previous commentary and it looks like this was going to be an accepted translation, but it still isn't. The position of the adverb there may seem odd to a non-native speaker, but it sounds fine to me.
I am as confused as others about this one. "What have you done with him there?" sounds very strange. On the other hand, both "What have you done with him (without "there") and "What DID you DO with him there" sound fine. If I get this one again, I'll try "What did you do with him there" and see what happens.... I think it might work with "udělat," unlike "What were you doing with him there." But we'll see!
UPDATE -- Well "what did you do THERE WITH him" was corrected to "What did you do TO him THERE," so I'm still confused. And now I'm confused about both the placement of "there," and about why "with him" was corrected to "to him" this time around. On the bright side, "what did you do" looks like it would be an acceptable alternative to "what have you done."
I think I can see why you might want to have the past simple case here. It is not a topic in which I am too strong so I intentionally did not comment the previous one. But my understanding is that the there expresses some specific occasion and so the past simple tense is in order.
We do accept the past simple case, but I have just spotted an error which prevents accepting the past simple with "to him". I will correct that.
But I will leave changing the default to others. At least now.
I'm not sure I will explain this well, but here goes...
I would say that "What did you do there with him" is more natural and straightforward, which may be why it's being reported so often. It pretty much means both he and I were "there" and someone is simply asking what the two of us did while we were "there."
On the other hand, "What did you do with him there" could have two potential meanings: (1) As described above and (2) While both he and I were "there," someone is asking what I/we did, considering that "he" was also there (and the person asking knows that I may have preferred to be "there" alone or with someone other than "him"). But meaning 2 is totally contextual, so I'm not sure whether this is helpful or not!
There are two problems with "What have you done with him there". First, the position of "there" seems rather awkward. Second, past tense should be used when talking about an action completed in the past.
I'm referring to the exercise where you have to pick words from a given list. I agree with BoneheadBass, "What did you do there with him" is more natural and straightforward. Please fix.
"What did you do there with him?" is now an accepted translation. Since I'm not a native Czech speaker, and not a sufficiently advanced student, I'm not in the position to decide whether the main translation, which uses "have you done," actually should be changed.