lumen is light, and when it comes to light in the room and how dark it have to be before you get a good picture on the screen.
the more lumen a projector have the brighter the picture will be on the screen ( witch will give poor black on black )
you are locked to 50/60 HZ/FPS and a higher latency compare to the monitor.
and by gaming they aim at console players.
as a pc games you have to look at lag from PC -> to the picture is shown on the equipment you want to use.
the higher HZ the bether it is and less screen tear you get.
resolution and picture quality.
the majority of TV's for example, are "listed" whit "400+HZ" witch is just bull, it take that frame and post it several times until next frame comes, this just to smooth it out.
the reality is that when you connect a PC for example to a TV ( whit what ever connection type ) it will stay at 60HZ no matter what.
there is TV's that is coming out whit 120HZ in but they are HDMI 2.0 and top of the like, aka expensive.