Could the Green party and Democrats form a coalition and choose the President accordingly

They certainly could, but why would they? Not only democratic party stand for a lot of things greens find unacceptable (and vice versa), but disproportion between both parties is so huge that greens would at best got given some paltry compensation (and a huge bill of firming democratic party atrocities with their names, this would essentially be their end) and most likely just become completely ignored and cut off after election.