At 6 feet away, a 52" television is far more than enough to still appreciate a 1920x1080 resolution over a 1600x900 resolution, so on that front you're covered. Also, 6 feet away should be just enough to take in all (or at least most) of the screen without having to move your head or eyes too much. So with that you should also be fine.
I'm not sure what you mean by "how does a PC port 1600 x 900 (roughly) on a 1080p tv if the tv is a 1920 x 1080 and my pc also can output at that res?" If you're asking how does a 1920x1080 television display a 1600x900 image, well then think about how you view youtube videos. If you're watching a 480p video on a 1600x900 monitor, and you put it to full screen without changing resolution, the image will take up the entirety of your screen, but your monitor is still 1600x900 and the image is still rendered at 480p (853x480). The image is just interpolated, which basically means that your computer performs a series of algorithms to scale the image to fit a larger screen resolution. The same thing would happen if you put a 1600x900 image onto your 1920x1080 screen. However, I'm not sure why you'd be concerned about that at all. Just have your PC start rendering at 1920x1080 and then you won't have to worry about scaling and you'll get a nicer looking image.