I currently use a setup similar to what you're going for. I'd say as long as the monitor supports HDMI and 1080p, you should be fine visually. Not all games run at 1080p, but for those that do you'll get the best image possible.
The Xbox 360 does support audio over HDMI, but you can opt to use either RCA or digital audio instead if you want. All of the manuals for Xbox 360 hardware are available at http://support.xbox.com/support/en/us/nxe/hardware/manuals/InstructionManuals.aspx (although I don't see a manual for the new Xbox 360 S that was released a few months ago, but I'd guess that as far as you're concerned the Elite manual should give you the info that you need). The manuals go over the various combinations of audio and video that are supported, so hopefully that would be able to answer your questions.
For speakers, I'm currently using a 2.1 Logitech system that I've had for a long time. They meet my needs, but I'm no audiophile. I'd suggest not using your monitor's built-in speakers, since those types of speakers tend to not be very good.
I also have a Turtle Beach headset which is much better than the headset shipped with the console - both game audio and voice are sent to it, and it auto-balances the sounds during game play so that voice chat doesn't get lost among the game's audio. Since the headset is wireless, it requires two AAA batteries, but I use rechargeable batteries and they last for a long time.
As for controllers, the only time I've noticed a loss of signal is when the batteries are getting low (but then again, I'm sitting at my desk only a few feet away from the console). I've also noticed that battery life is quite good, but I use standard rechargeable AA batteries rather than the Microsoft battery pack.
Based on a quick search, unfortunately it looks like it won't. While it seems like it should work (due to having 4GB of internal flash memory), it doesn't probably because the original Xbox didn't know about flash memory, only its internal hard drive. Since really running original Xbox games on the 360 boots up the equivalent of an original Xbox, that's all it can write to.
Buying the 250GB hard drive (or, better yet, just buying the model that comes with the hard drive - it's cheaper that way), should do the trick and allow original Xbox games to run on the Xbox 360 (well, those that have been updated to work anyway).
Best Answer
You can use component cables to play Gears 3 in 3D. I just did on my own HDMI-less console. You might get a better image quality with the HDMI, but the 3D affect does work with component cables.