1) Invasion as in the Lingsham incident or the attack on New York? For New York, there never was an invasion. It all started with the Ceph virus, in the middle of that chaos alien ships appeared and they started attacking. At the end you come to know that the alien ceph were based under New York all along (probably explaining their origins and why they were different in appearance to the lingsham aliens)
2) The Nano suit team were earlier under the US government (Crysis 1 and Warhead) although in this game they seem to be reporting to Crynet (mainly Hargrives) ... although Prophet went AWOL.
3) As Hargrives said, the nature of the aliens were to take back their land. The aliens were a prehistoric species.
Although I agree with you that it is a bit disappointing and vague to not hear a word from any of the main characters of the original Crysis. Although my guess is that they died because of what Prophet said - "We (as in Prophet, Nomad, Psycho....) were dead men walking". So perhaps they were all infected and dying anyways.
Or maybe we're going to have a kick-ass Nomad-Psycho-Alcatraz Crysis 3