For the football you don't need a darkened room or a grey screen. The content is bright. There's not a lot of shadow detail to pull out from the picture. I can say this because I go far enough back to the days when three lens CRT projectors were how we watched football in pubs. They were nowhere near as bright as digital gaming/entertainment projectors. Those Seleco, Vidikron, Barco, Sony and other CRT projectors were the equivalent of a 600-800 ANSI lumen digital back in the day.
If you're gaming, and it's something like a night ops FPS, then you might want to look at the light control in the room down to a point. That last part is important. Stick your projector on after sunset. Make sure it's on the picture mode you use for regular dark room viewing. Fire an image at the screen/wall but with nothing playing so you're projecting black, or use a film with large black borders. If you can see those borders are brighter than the room's background darkness level, then the projector is already projecting black at a brighter level than your current room blackout. Unless the projector brightness level is poorly set, then the limiting factor is the projector rather than the room.
This is a situation where a test disc comes in really handy. Brightness needs to be set for the room conditions. If you put up a brightness test pattern, and then find that reducing the brightness setting starts to block out the above black bars at 5% and 2% before you can make the projector's black as dark as the room, then you know you've reached the limits of the projector.
A grey screen will help with the black level, but it can obscure shadow detail in the process. That's not good. It's possible to get some of the shadow detail back by changing the Gamma setting, but that might have consequences for the colour saturation and grey scale tracking. So it's all a bit of a balancing act and a little bit of a compromise.
Speaking of compromises, one has to think of the screen surface itself. High-end grey surface screens are an engineered product. That means they have properties beyond just the colour of the surface. These properties give them advantages over basic grey surfaces such as better focus, improved colour accuracy, less colour-shift, reduced hot spotting/greater image uniformity and better ambient light rejection. The rub is cost. Grey screens worth owning aren't cheap. Stewart and Da-Lite brands will set you back the thick end of £2K+ for a motorised 8ft. That's just not sensible with a £500 projector.
Budget grey screens are not engineered in anything like the same way. The stuff at £80-£300 depending on size is little more than pigment tinted PVC. Over all, if buget is a major consideration, then I'd be more confident buying a basic matt white as there's less chance of the manufacturers getting that wrong.
The bottom line is
