Puritan Board Junior
Yes, but that is pure speculation, and one would be very hard pressed to prove that any of Nestorius' writings were post-Chalcedon. Granted not impossible, but highly unlikely. Moreover, Nestorius was not at that Council, and it's not as though he was living in the days of electronic communication of documents and conciliar decrees.
No, but he was living in the days of the imperal postal service, which if I recall my source correctly, could in the Roman days of a century or so earlier, transmit a despatch from Rome to Palestine in well under a month. Unless you know of a substantial degredation in the imperial postal system in the east, I don't think we can presume anything substantially longer for the news of Chalcedon to spread empire wide.