Not necessarily, most tv's should have Component (Red, Green, Blue, Red, White, Yellow inputs) as well, and all monitors should have a VGA port. Both of these are HD compatible.
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Yes, any HDMI cable can work on any console, including the Xbox 360.
Yes.
no you cannot
In order to get High Definition gaming You need an HDMI cable and a HDTV/ HD Computer monitor.
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.