A 50" or 60" diagonal is very small for a projector image. 50" diagonal 16:9 image is 43.6"width x 24.5"height = 7.4 sq ft. 60" diagonal 16:9 image is 52.3"width x 29.4" = 10.7 sq ft. Image brightness is projector lumens devided by image area in sq ft multiplied by screen gain (1 for matt white). A projector with say 800 lumens on a 50" diagonal 16:9 screen would be 108.1ftL, and on a 60" diagonal 16:9 screen would be 74.8 ftL. A flat screen is usually around 30-40ftL and designed to be watchable with the living room lights on. Projection screens are usually aimed at 12ftL to 16ftL with the room lights off, with some people prefering 8ftL in a batcave and others upto 24ftL with some ambient lighting. You are going to need to use a low gain greyscreen which will also help reduce washout of simultaneous contrast in a non-dark painted room, or/and a camera lens neutral density filter to drop the brightness. A nd2 filter halfs brightness, a nd4 quarters it. A very bright dlp projector image maybe unwatchable due to dlp rainbow effect, while a very bright image by other projectors will just give eyestrain with prolonged viewing.
When calculating screen brightness, I would recommend treating the manufactures quoted figures with a pinch of salt. Using low lamp mode to increase lamp life and after a few hundred hours of lamp, not using white peaking or brilliant color on a dlp projector, and calibrated for accurate colour, you maybe looking at 50% of the manufactures claimed lumens. Lamps dim by upto about 20% in the first few hundred hours then slowly down to as low as 50% of their original lumens during the rest of their life.
Many lcd projectors and very few dlp projectors have vertical and horizontal lens shift. Projectors also have digital keystone correction for non-central mounting. But for best picture quality it is generally advisable to not use much or any lens shift and no digital keystone correction. So the projector should be mounted (left/right) in line with the middle of the screen and mounted up/down according to the projectors lens offset.
Distance devided by throw ratio equals image width (not diagonal). For example with a 8ft width image and a projector with a throw ratio of 1.5. The projector lens needs to be 8ft x 1.5 = 12ft distance from the screen. Projectors quote a range for throw ratio as they have some lens zoom. Typically arround 1.2x. If you project the smallest image at the distance you get higher contrast but lower lumens, if you project the largest image at the distance you get lower contrast but higher lumens. If you set the zoom lens to the middle for the middle sized image you may get a slightly sharper - more infocus image, depending on lens quality. Since image brightness = projector lumens devided by image surface area in square feet, the differences in image size more than offsets the differences in lumens, so the larger image will be less bright because it is larger and the smaller image will be brighter because it is smaller.
To calculate image offset you multiply the images height by the offset. For example with a 54" tall image and a offset of 125%. With the projector on top of a table top the center of the projector lens would be 25% of the image height below the start of the bottom of the image, so 13.5" below the bottom of the 54" tall image. With the projector on the ceiling (upside down) the center of the projector lens would be 25% of the image height above the start of the top of the image, so 13.5" above the top of the 54" tall image.
For the screen you can use a white matt painted wall, I use dulux light and space absolute white rich matt with lumitec technology for added brightness, but I have a very big screen in a batcave, a diy low gain grey paint mix may suit you better, some websites list paint mixes for diy screens. You could get a low gain commercial screen. My only reservation about commercial screens, is they are fabric and at your probably very short viewing distance ( viewing distance of 1.5 x image height is the average for projectors) you may be able to see the texture of the fabric, so I would recommend getting samples to see how smooth they are if you go that route.
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