The EDID (info passed from TV to the computer) many times will not function correctly from HD TV's to computers. The 'handshake' at this time still has no clear standards.
In System Preferences - display; only option you have is Detect Display, but many TV's will not pass their info along unless they are awake and ready to pass info during the boot process.
Many will not work unless you turn on the TV prior to booting the computer.
(Based on my own experiments and much reading of others trial and error over last couple of years. Some manuals are now stating this also. Turning on the TV and then the computer, DVI to HDMI should work with most TV's in XP and OS X. It still will not work in any Linux that I've tried or heard of so far.)
If this does not work for you, I would suggest next step would be to e-mail Sharp tech support. If you call in and get someone that has an answer for connecting a computer, I would be very surprised. We are still a young breed
In reference to the display resolutions listed, don't worry about that. The one you are seeing is specifically for your MBP. Once you get both screens up and open up Displays - you will have two of these windows, one on each screen. Each window only adjusts the screen it is located on.