VideoCommon: Header cleanup

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

Source/Core/Core/State.cpp

@@ -10,8 +10,10 @@
 #include <vector>
 #include <lzo/lzo1x.h>
 
+#include "Common/ChunkFile.h"
 #include "Common/CommonTypes.h"
 #include "Common/Event.h"
+#include "Common/FileUtil.h"
 #include "Common/MsgHandler.h"
 #include "Common/ScopeGuard.h"
 #include "Common/StringUtil.h"