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The display issue in MilkShape is caused by the close proximity of the two sides of the cover. They are two different sets of faces, one turned up and one turned down. Because of minor variations in the coordinates, in some locations the bottom side is above the top side.
MilkShape has a display option "Draw Backfaces" that you access (for each window) via a right-click menu. In your case, you have that option selected, and the bottom-side faces that appear above the topside are drawn in black. If you turn that option off, those faces will be invisible (from the topside).
I am not sure if this is the problem with the bed in-game or not. The game renders backfaces as invisible, so it should render like that.
If you like to say what you think, be sure you know which to do first.