No, they didn't.

The didn't because the Roman government wasn't the kind of institution that produced those kinds of documents. Even in the early imperial period the state 'body' was in many ways just an extension of the emperor's vast personal 'household'. There didn't exist the kind of 'government' that produced various documents of the type, that, say a modern government would. The Roman 'state' produced decrees.

At times, though relatively limited times, those decrees proscribed Christian belief and practice, but they didn't do so with reference to debunking belief in the death and resurrection.

Critiques of Christianity in late antiquity are generally written by individual authors from a philosophical standpoint. Celsus, for example, and his late 2nd century "The True Account", which was one of the first sustained written critiques of Christian thought and belief.

No, because that claim (as far as "history" is concerned) did not arise until far later, and well after a body could be produced (if there ever was a body).

It was not, as far as history is concerned, a contemporaneous claim that could be disputed. There is no evidence of Jesus at the time of his life, and so there will necessarily be no evidence of reaction against him at the time of his life. Frankly, it would be the most important historical find of Western civilization if we found a contemporaneous Roman attempt to debunk Christ's resurrection, given that we can't even find a friendly one that might be biased!

If someone claims that Lincoln rose from the grave now, in 2016, one would not expect to find evidence disputing that claim among the papers of Confederate generals. New claims cannot be expected to find counterclaims in old evidence.

Even if the New Testament is accepted at face value as accurate history, the Book of Acts says that the disciples never told anybody about the resurrection until 40 days after the crucifixion, by which point no body would have been identifiable anyway.

Looking at purely historical evidence, is no evidence that the Romans were even aware of such a claim for many decades after the fact. There is actually no evidence that anybody ever made a claim about a missing body until Mark's Gospel around 70 CE. Paul says Jesus was "seen" by people after his death (including by Paul himself), but never says anything about an empty tomb or a missing body, or describes what anybody saw or claims that the disciples saw Jesus in a different way than Paul did. Everything Paul says (especially in 1 Corinthians 15) is consistent with the original claims only have been derived from visions or personal revelations like Paul's own.

Even if the Roman authorities had been made aware of such claim, there is no reason to believe they would have cared or tried to debunk it any more than the US government would bother to try to debunk internet rumors that David Koresh had "appeared" to people.

As a follow up question: How did the different sects of Christianity interpret the life and death of Jesus? I understand it that they're were a few, such as Roman Catholic, Greek traditions before the Schism, as well as Arian Christianity.