Sorry I meant to say sorry she's getting the TV
There are some specially designed projector screen paints, but you're still going to want to have a dark room to use them. If you paint the wall black, you're going to have a hard time getting a projector bright enough for it to look white again when you want something bright.
Check out the screen size calculator function here
(The little calculator button next to Throw Dist)
Making the image smaller will increase the brightness since it only has to light up a smaller more concentrated area of the wall.
With a smaller screen, artifacts, screen pixel structure, etc. will be less noticeable, but you'll be giving up size which is the primary reason to get a projector with its cinematic feel.