about to buy Xbox360, questions

I've got a couple questions for you all.

can I hook up my xbox wirelessly over the internet so I can play live?

do I need any other cables? hdmi cables? for better picture.

thanks.

Wireless adapter is £50/$100 or you can do it with wires for free.

HDMI cable comes with elite and component cables with the pro with standard cables with the arcade. So you don't need to buy cables for HD with the pro or elite.

I have lynxs wireless router i just baught. that should work, correct?

also, I do have a 1080p TV (50 inch). Would ity make sense for me to buy the HDMI cables then? I thought the PS3 was only able to display 1080p.

thanks for the help