Receiver w/ Surround sound?
If you have planning on having an A/V receiver for surround sound in the main room, you could use it to switch multiple component video, svideo, and composite video sources to a single component video output. If you purchase a more expensive models, you can also get HDMI switching and connect the HDMI output to your new TV and the component video output to the 2nd TV without the need for splitter and amplifier boxes. Don't forget, you will also need to get audio to the 2nd TV via a digital (coax or optical) or analog connection.
Yes, RG-6 with RCA connectors would work great for component video, digital coax audio, or analog audio cables.
With your new TV, you will want at least a progressive scan (480p) player. You might also see some benefit from an upscalling version but if the scaler in the DVD player is better than the one in your TV. For one of these, check out the Oppo Digital for $200.