VideoCommon: Header cleanup

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

Source/Core/VideoBackends/Software/SWRenderer.cpp

@@ -8,10 +8,11 @@
 #include <string>
 
 #include "Common/CommonTypes.h"
+#include "Common/FileUtil.h"
 #include "Common/StringUtil.h"
+#include "Common/Logging/Log.h"
 
 #include "Core/ConfigManager.h"
-#include "Core/Core.h"
 #include "Core/HW/Memmap.h"
 
 #include "VideoBackends/Software/EfbCopy.h"