I want to know whether I'm actually playing in HD with my 360 and my monitor.
I'm using an LG widescreen monitor, usual res. of 1440x990, but I don't know about the res. it's set at for my 360. I'm using an official VGA cable plugged into either a VGA port or DVI port.
Would that be capable of displaying HD? If not, would I need a new monitor with an HDMI port? Or could I get some kind of DVI-HDMI cable?
Need to check what resolution your Xbox is outputting at - It'll be in the Xbox's display settings someplace. With the VGA cable I'd imagine it defaults to the highest available resolution of the monitor, so you'll likely be effectively playing somewhere inbetween 720 and 1080, though in all honesty, I wouldn't worry too much about what it's set to - you likely won't see much of a difference in the higher modes, at the size of your monitor. (unless it's a huge monitor with a ridiculously low resolution panel, for some reason)
I doubt there's much of an advantage in switching to DVI or HDMI. HDMI can carry audio, but most monitors won't have any way of outputting it.