If you want to game and not lose any performance you can simply SWITCH your screen to the HDTV leaving the laptop screen unused.
You probably will have to do an initial setup of the screen. That's in the control panel, HDTV section and you'd choose 1080p_NTSC60 or something nearly identical (1080p, NTSC, 60Hz is the best your HDTV supports). You may need to adjust overscan as well.
Some laptops also have Intel WiDi which you can Google if you have it.
Sound should works as well, but if for some reason it doesn't automatically switch, you may have to manually change the "Playback Device" to HDMI (right-click the AUDIO ICON in the lower-right of desktop).
There are NO STREAMING issues or lag. The only performance issue might be having both screens running which would only matter for games. I have no idea what difference that would make.