Cutscenes

If I manually close the GW client during the final cutscene of a mission, when I log back in will that mission be "completed" or will I have to start all over?

Did you see any shields and swords in the cut scenes or if you are in Eotn a burning tapestry. If not you might have not completed the mission.

I remember a very long time ago me and a friend were doing the Wilds. We were going through the end cinematic when our internet died (we were both in the same house) and we err7ed. We hadn't seen the shields yet so when we logged back in we had to redo the mission.

I can only assume it's the same way now.

I don't know about Prophecies, but at least in Factions and Nightfall you can see the mission completion rewards appear in your chat window right before the cutscene starts. Those missions will count as completed even if you exit Guild Wars right after the cutscene started.