Well you have picked a 16:9 screen and a very fine projector that has functions to be able to be programmed for any method of presentation you would like.
Before deciding on seating distance you should select the method of presentation that suits you best. You mentioned movies and games and of course you want your seating distance optimized for listening so speaker placement will be a function of seating distance for the music.
Starting with movies they come in all kinds of aspect ratios and types and each have their cinematography based around the assumed presentation they would be given in commercial theaters. The three popular methods right now are scope, flat, and IMAX. Old movies are Academy and new movies made for streaming TV like Netflix are experimenting with new AR like 2.0:1.
Flat and Scope are intended to be shown the same height where the scope movie is more immersive as it is wider and people that mainly watch them don’t have 16:9 screens they have scope screens 2.4:1. This presentation is called CIH constant image height.
With the advent of IMAX now more mainstream on home media with movies like Dunkirk and Aquman in part in that format and a push being put on releasing more IMAX versions of other movies (search IMAX Enhanced) those presentations should be at least as wide as scope and taller than CIH. Some people refer to this presentation as CIH+IMAX. It is roughly what I do.
To answer your question for myself with your equipment I would watch Flat and scope movies 1.85 & 2.35 at a distance equal to 2 times the image height and I would watch IMAX movies at a distance of 1.5 times image height. I would do this with programming the zoom, focus & shift to adjust the image size and location on a 16:9 screen. 135” screen @ 8.25” or 150” screen @ 9.25”. That would be a CIH+IMAX presentation that would suit me.