I'm curious to know if any diplomacy was ever conducted between the Axis and Allied powers after war was declared. Was it just assumed that talks would start in the event of one side wanting to surrender or was there a constant dialogue during the period of conflict?
I can only address one portion of your question, specifically backchannel communications through a third party.
One backchannel of communication that was used was the Vatican. It sounds dramatic, but the reality was not as cloak and dagger as it sounds. Largely, the communications were unofficial and asynchronous. So far as I know there were no meetings of combatants in the Vatican. Rather, each side would approach a Vatican official with a message for their counterparts which would then be passed along--usually verbally rather than in written form. The most famous of these was a group of German officers approaching the Vatican to feel out British opinion on what would happen if there were a coup that overthrew Hitler. This first occurred in 1939-40, and was repeated in 1944. The communications were vague, and nothing came of them.