"...until my task is done..."

[HerenIstarion]

I wonder...did Gandalf's possession of one of the three elven rings of power (Narya) when he fell have anything to do with the reason he was sent back

I'd say it's the other way round - the possesion of the ring was the consequence of his main task - it was helping him to encourage free people. Quote by Cirdan when he was handing down the ring may come in handy, if anyone can paste it here. It was something along the lines - "you need it more than me, and it will awaken fire of hearts"