Though perhaps thought spurious, due to the fact that - “No Governor of Jerusalem; no Procurator of Judea is known to have been called Lentulus.” But Lentulus was a consul it appears, so he need not be either a notable Governor nor Procurator within the province to provide such information to Tiberias - as the consul (likely a suffect consul) the office would have been held for perhaps a half years time in those days.
Yes, but that’s not the only reason it’s thought to be spurious. There are other points that weigh against it, such as expressions in the writing that would not be used by a Roman, the complete lack of provenance of the letter, and the very late date of “discovery”, which coincided with a period in which forgeries of this type were very common.
Material that does not survive in manuscript form is, even in the best of states, could only be put forward extremely loosely as “from the Roman archives.” That such a notable alleged manuscript might be simultaneously overlooked, lavishly bound, and conveniently lost taxes the limits of credibility beyond at least my own breaking point.