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
Is your second controller synced to your Xbox 360? If this controller is not synced the console will not take inputs from it and the "Start" button cannot be used to load a second player into the game.
Refer to this guide on how to Sync your Xbox 360 controller
Alternatively, you may have reached the maximum amount of players on your Minecraft world. On Xbox 360 there is an 8 player limit on a Minecraft multiplayer world. If you attempt to exceed this by adding a 9th player via split-screen you may be prevented from having them join.
If that doesn't work try restarting your game or console.