1) Rumple gave Regina the mirror to send Cora to Wonderland. So he always knew where she was. Sending the letter to her probably wouldn’t have been a problem.
2) I think Rumple was just so angry with Cora for dishonoring their agreement and its worse because they had a deep relationship that failed so he was hurt and angry + plus more angry because nobody breaks a deal with Rumple. His anger for Cora probably just drove him to write the letter to Cora to upset her. That he wins because she tried to screw him out of the deal and he found her first born regardless of her deceit.
3) I don’t think Cora tried to get in touch with Zelena during her training. Cora had been stuck in Wonderland and needed to imprison the mad hatter to find a way to get back to her land. I don’t even know if Cora remembers Zelena. Maybe on top of ripping out her own heart so that she won’t feel pain she also took a memory forgetting potion like Regina.