would connecting a hd lcd tv to your computer and setting the video card to hd chew up all the memory?

Answer 1

No.

Computer resolution have been higher than HD TV resolutions for quite a while. 1080 is a 2M pixel resolution. 720 us <1M pixel. - 30% lower than a 17" LCD panel

The higher the resolution the more memory the graphics card will need for each memory map it uses, and the more memory it will need for calculations but I doubt it would make a significant change to the amount of shared memory assigned to video. If it is a separate video card then the memory is probably not shared in the first place, so it will make no odds.