Nowhere in the Gospel are we told who wrote it. It has, however, always, from the beginning, been associated with Matthew. The reasons to doubt it aren't good enough to overcome that one piece of evidence. As to when it was written, the questions raised have only to do with a couple of things: prophecy about the destruction of the temple but that is only an issue if you don't believe it's prophetic, and the idea that Matthew is playing out his personal beefs with the establishment Judaism. Early dating seems far more likely.
