Okay,
Он вышел из дома, отправив письмо ----- He left the house after he had sent the letter or He left the house after sending the letter (отправив --- completed action already)
Он выходил/выходит/будет выходить из дома, отправляя письмо ------ He was/is/will be/ leaving the house while sending the letter (two simultaneous actions) (отправляя --- the action is still in the proccess)
Он выходил из дома, отправляв письмо ------ He was leaving the house while sending the letter (two simultaneous actions as well) (отправляв --- the action was still in the proccess at some moment in the past)

About the last example, I don't even think one can use the word отправляв, because I'm not sure it's the word at all. Он выходил из дома, отправляв письмо just sounds really strange. I personally would say Он выходил из дома, отправляя письмо.

So I just showed you what the word отправляв would be if it's a word.