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.
Best Answer
It's possible to do this electrically, although mechanically the two cables interfere. Microsoft sells a special version of the AV adapter that you can use to plug both cables in at the same time, or you can break the outer casing on the AV cable so that you can fit the HDMI cable in at the same time.
Note that plugging both cables in at the same time might cause some issues if you plan on switching between inputs on the monitor while your Xbox 360 is still on, see:
Why does my Xbox 360 reset when I change the input on my TV, unplug the HDMI cable, or turn the TV off?