VideoCommon: Header cleanup

Also remedies places where the video backends and core rely on things
being indirectly included.

Source/Core/VideoCommon/VideoBackendBase.cpp

@@ -2,6 +2,9 @@
 // Licensed under GPLv2+
 // Refer to the license.txt file included.
 
+#include <string>
+#include <vector>
+
 // TODO: ugly
 #ifdef _WIN32
 #include "VideoBackends/D3D/VideoBackend.h"