For gaming, the new sony 285 line (or better, 385, 675) is perfect. It has a low latency game mode and true 4K. I use my theater primarily for gaming. I'm 50% gaming, 24% music, 24% tv, 2% movies. After playing games on a 135" screen, you will never go back to a small TV or PC monitor for gaming. And 4K games looks tons better on a 4K projector than 4K movies.
The last 2 JVC projector lines also have low latency gaming modes, but those are not true 4K.
Video: JVC RS640 135" screen in pure black room no light, htpc nvidia 1080ti.
Audio: Anthem mrx720 running 7.1.4, McIntosh MC-303, MC-152, B&W 802d3 LR, B&W HTM1D3 center, B&W 805d3 surround, B&W 702S2 rear, B&W 706s2 x 4 shelf mounted for atmos, 2 sub arrays both infinite baffle: 4x15 fi audio running on behringer ep4000 + 4x12 fi audio running on 2nd ep4000.