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Usually i run in dual display ... and also ive noticed you cant close the lid on most laptops when running an external monitor since it will tell the laptop to shut the display off. try restarting the laptop after having the tv hooked up its also possible the cable you got is bad ..sometimes on those cheaper RCA plugs the wires break inside.
A good way to tell if the connection is working is that most systems when they are showing the post screen when you turn them on will display all the video outs so you can still see stuff on the TV, if that doesn't happen when you turn the laptop on then there is something wrong with the connection, if it does happen but when windows boots the screen goes off then you have an issue with drivers not sending a signal to the tv.
Another possibility is that the AV input on the TV is bad, try the diff inputs on the TV and see and make sure you have the video connected to the correct input.
as for resolutions ... when runing TV from PC you can run between 640x480 to 1024x768 (sometimes even 1280x1024) it really doesn't matter for video watching. i'd just pick whichever one best fills up your TV's screen.
btw, 720P is 1280x720, 1080P is 1920x1080, screens are available in various resolutions but they implement some sort of scaling up/down scale the video to fit the panels native resolution. In the end the native resolution of the screen doesn't matter O_o
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