For an overview of planning a project, see "Planning your work".
First, find out what the dimensions (in pixels) the projected movie needs to be. 1920x1080? 720x480? Something else? Whatever the answer, make your composition dimensions match the required dimensions for the output.
Next, find out what is going to be used to play the movie. You mentioned QuickTime Player. So, you'll probably want to render and export to a QuickTime (.mov) container. The H.264 video codec has pretty good compression to keep your file size down, so that might work. Try it.
I'll repeat that last sentence: Try it.
Render and export test movies and try them on the exact equipment that will be used on the big day. That is by far the best way to be sure. It eliminates all guesswork and speculation. You don't need to test with your finished movie. You can test with something representative, like an animated test pattern or a piece of stock footage that you have transcoded to your required specifications.
I tried redering with the H.264 format, and the file size was good, but when played in quicktime, it was just a black screen. I tried it with another comp, and it was pixelized and the colors were completely off.
Then try something else.
Determine the native resolution of your projector. Make sure the playback computer is set to that resolution. Render your content to the same resolution.
As mentioned in your other thread, please post more information about your comp settings and output settings for an assessment of your output problems.
To add to what the other folks have said, the limitations of your playback device determine the size of your AE comp.
Does this projector have a limitation on size, or a preferred size? You need to find out.
Does the playback software have a specified or preferred codec? You need to find that out.
The size of the screen a projector uses has no bearing on the size of your comp. Think about this: you can watch a movie on a normal hand-held DVD player. You can play that same DVD through a projector, and see it on the side of a house. In either case, the DVD hasn't changed one bit: it's still 720x480 video.