I too have been looking into this the past week!
I need to move flat again, and going to get a new screen... and thinking that I'd quite like a projector this time rather than a telly.
I did simple google search for something like 'best projectors for gaming', and most were hyping the BenQ and the Optoma GT1080.
I then watched a fair few YT demos and reviews of these.
So far, I am leaning towards the Optoma GT1080.
It only has HDMI connectivity, but I too will be using my Nvidia Shield TV, Xbone, gaming PC ... and I guess perhaps some retro consoles via some adapters/
The main points leading me to this model;
-Very short throw, it can produce 100" display at only 1m away from the wall. It's probably one of the best for using in a smaller room, which will likely be me.
-Decent display, even with slight ambient light. Obviously the darker the room the better, but I think this looks ok in a room with a lamp on, from what I see on YT demo vid.
-Has a 'Gaming Mode', which apparently makes a great difference when playing games through it, improving f̶r̶a̶m̶e̶ ̶r̶a̶t̶e response time.
-For the performance, it's priced pretty well.
I suspect that I wouldn't watch too much telly on it, though perhaps a little, connecting the Tivo box to it for that.
Mainly, will be gaming or watching films via the Shield.
The short throw is most likely going to be the driver for my choice - Be interested in anyone has any other suggestions on recommended projectors too, particularly for gaming with
edit: corrected some mis info about the Gaming Mode, mentioning frame-rate rather than response-time