Screen size should not be decided until the projector has been used for a few weeks on a wall.
User preference is the ultimate factor, a very general ratio is 10-12" in diagonal for every foot in distance. That would be between 100-120". But there is a user who likes to sit at 7' with an 135" screen, 4K display.
You might want a big screen, which would limit the projector selection.
With such a large screen you will very likely want to use the setup for movies/TV.
While visual acuity differs, 100" from 10' might be unnoticeable with a 4K display.
Resolution wise, native 4K are amongst the sharpest. 4K DLP pixel shifters can also be very sharp, with some models being sharper than native 4K, but they're outside the budget. Native 4K and DLP 4K pixel shifter have ~8m pixels, while Epson's 3LCD only has ~4m.
Other components such as lenses have an impact on sharpness, not just the resolution output.
The source is also important. Many movies are actually shot in 2K and repacked at 4K. There is a difference between a title shot in 4K. With video games it can get complicated since the internals can be other than 4K, especially on consoles.
4K projectors can do an ok job with 1080p upscaled to 4K, but a 4K source is where the improvement will be seen.
What is the gaming platform, next gen XBOX only?
UST projectors are rather expensive for what they provide, you're partially paying for aesthetics. As other have mentioned above, UST ALR's are not the same as regular throw ALR's.
Lumen values should not be taken from next to the product name. There are some reviews that measure each preset in each lamp mode, like Projector Central reviews.
DLP 4K when inputted 1080p can vary from 40ms to 80. There has been some suggestion that with a 4K signal the lag is lower, but no one has measured them yet. The TK800M and HT3550 sit around 40-50ms lag with 1080p.
Native 4K latest generation JVC are low 30's and Sony with high 20's. With a 4K signal, maybe even with 1080p, can't remember right now.
3LCD on Epson is low 20's with 4K and high 20's with 1080p.
The issue with Epson is their motion handling is not the best, but they are still used for gaming. At the bottom of the page there will be two videos between a 4K DLP and 4K 3LCD. In the second video the right side is quicker because the Epson (on the right) has lower lag.
DLP has the best motion handling, followed by 4K native JVC and Sony, with 3LCD being last.
suggested even a white screen can be used if the lights are controlled, not too bight, don't hit the screen directly, etc.
While you may not notice, some ALR screens might introduce visual artifacts. Some visual artifacts appear if the projector is placed to close, and while it differs from fabric to fabric, the recommended throw is somewhere around 1.8x or above. The throw is calculated with the screen width.
Some projectors don't have the necessary throw for using ALR screens.
Some examples of how angular long throw ALR screens behave:
ALR screens can be built for very little, average performing ALR fabrics can be had for under $200. If you can build a wooden frame, that would be very budget friendly:
Here is an example of a screen with Carl's ALR:
Paint is a nice option for a screen that improves black levels, with ALR properties. It's inexpensive and can be applied on a cheap white PVC screen. The advantage is it can be customized to your setup.
As others have also suggested, getting an inexpensive budget projector first to get an idea of what you might want for a projector is a great idea. The Viewsonic mentioned above has ~16ms lag. The Benq HT2050A is a very similar model with the same amount of lag. They both have a RGBRGB color wheel, which has the best picture quality (from the selection of color wheels).
The first and best suggestion is to get a budget projector and test it out. There are 4K DLP projectors with low lag in the works, but not available for purchase right now.
A few other options:
Native 4K Sony/JVC and a white or ALR screen. B stock JVC NX5 is about the same price of an Epson 6050UB or lower. PQ wise and gaming would be the best option here. It's great black levels, as well as black levels of other projectors will be affected by ambient light. So for gaming with ambient lights this aspect will be negatively impacted, but a 4K native projector with poorer blacks (still ok) is still around the same price. The nice blacks would be a benefit for lights off movie/TV.
4K Epson 5050UB. If the screen is not too large the ~4m pixels will not be obvious. The issue here might be motion handling. If you can, go see a 3LCD Epson live in a store, maybe compared to a 4K native or DLP to ascertain if motion handling is ok.
4K Epson HC3800/3200. Bright. The HC3800 has better blacks than the 3200, but if there will only be ambient light gaming the 3200 will be a better choice.
1080p laser like the Optoma ZH403. It's got ~32ms lag. Can accept a 4K signal. Budget friendly.