The campaigns can only be completed once. In the Vanguard playlist you will find heroic versions of campaign missions, so while you can't play the full campaign from start to finish, you can re-play missions at an elevated difficulty level via the Vanguard playlist. Each week there will be a random sampling of campaign missions available to play.
Regarding question #2, once you obtain an emblem it is always available and unlocked in your Collections. So I'm not sure why you're asking why you still have the emblems in the screenshot.
On question #3, if you hover over an emblem, many tell you the source of the emblem. If you wish to name a couple specific emblems you don't currently have then someone can reply in this thread as to how to get them if it isn't clear from inspecting the emblem.
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Thank you for replying swfan77, I appreciate that. Thanks for clarifying question 1. I believe it would be awesome if bungie would allow users to reset the campaigns quests so they can play the original campaigns again. Let me clarify question 2 and 3 As you can see in the screenshot I've got the first three emblems for completing the campaigns, but how am I supposed to get the emblems in the second row? I've played and finished all campaigns on all 3 characters two times but I didn't get these emblems...