I have played the level several times, but the flag in the main map always remains in grey color, so I can’t play new levels. Why?
As far as I know, the multiplayer part is not necessary for the singleplayer completion. Are you sure there is not another flag? I know the arrow points at the multiplayer one, but there should be another one. (Most likely one the roads.)
Try playing both as Team Red and as Team Blue.
A multiplayer-level is never necessary for progress in the single-player-campaign.
As Gundericus has mentioned you should try submitting as both teams (your second code doesn’t have to be anything meaningful, but see it as training).
There definitely should be another grey flag around. If not look for small purple arrows, they lead to new (or old) areas. Your newest level might be there.
I have submitted my score on both team blue and red, but didn’t find a grey flag n Backwoods nor the dungeon.
What do I do?
The level progression in Blackwoods must take you by the upper path above the castle and towards the right. Post a screenshot of your map screen.