Yes, any HDMI cable can work on any console, including the Xbox 360.
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Yes.
When the Xbox 360 first came out, there was no HDMI slot at all. After the release of the Elite (black Xbox 360), they introduced HDMI. After that, I believe all Xbox 360's was introduced with the slot. Basically, if you have the first Xbox 360 then no.
no you cannot
Yes, you can. All xbox 360 accessories (av cable, ethernet cable, memory unit, controllers, etc.) and games work with all versions of the xbox 360. The only exclusion I know is the HDMI cable which only works with Xbox 360s with built in HDMI port (all xbox 360s versions currently being sold have HDMI ports).
If your Xbox 360 gaming console has sound but no color then this could be a problem with your HDMI connection. Try using a different cable.