Yes, the Blu-Ray disc format is a high-definition one meaning that it is made for HD-TV's.
My suggestion is if you are thinking about gaming at all, get a PS3. It has a built in Blu-ray player that many agree is the best out there. It's only $399 which is much cheaper than the other Blu-Ray models out there. Good luck.
Yes. You will need an HDMI cable for best quality. Be sure to get Blu Ray as the competing format has now been dropped by all main manufacturers and only a few films are available.
Your confusion is probably caused by the consumer electronics industry's tendency to reuse terms such as HD which means High Definition for so many things these days. The bottom line is that HD-DVD and Blue Ray DVD were and are both high definition formats. Toshiba named their product HD-DVD and Sony named theirs Blue Ray DVD. Neither format is compatible with the other. And only players made to be compatible with both formats such as the only made by LG Electronics would play both of these disc formats plus the older DVD Video format and recordable discs based on that format. So the Blue Ray player will work with your HDTV. And how you connect it is up to the kinds of inputs that you have available. If you have HDMI input available on the HDTV use that for the best picture and sound (its all digital). But if you do not use the Component input along with its companion stereo audio inputs (5 RCA cables).