Yes, you sure can install a projector and screen in an apartment nicely. My recommendation is to not "just" paint (or use) the wall for a screen. That is a lot of wall, and you might want framed pictures up there. I think a pull-down screen is best for an apartment or any multi-purpose room.
A pull down screen with retro-reflective material, such as a High Power, can be mounted to your wall with few enough screws that you can fill the holes later without problem or objection. Just find the wood studs inside your wall to hold up your screen mount. I'm not confident that drywall anchors will be strong enough for the force of pulling down your screen every day, so do find where your wall studs are, even if you have to make a few extra holes along the way. (a stud finder usually works, though.) Mounting a pull-down screen is not that much more than hanging picture frames around your place. With a retro-reflective screen, your video images won't show the ripples that inevitably develop in pull-down screens.
You obviously cannot run wires through your walls or ceiling, so placing your projector on a shelf behind the viewers, along with your cable box, DVD player, PS3, etc. is your cleanest solution, as others have written above. That will require a projector with lens shift, which most lcd projectors have but few of the inexpensive dlp projectors have. Then, just run your speaker wires along your baseboard to your speaker locations. Sometimes you can jam the wires along the edge of wall-to-wall carpeting to hide them, and sometimes you might need to staple the wires against the baseboard. You can buy plastic tube like things to help hide the speaker wires if you have to run them straight up your wall to wall-mounted speakers, or you can just staple the wires to the wall, and fill the tiny holes that makes later. Try to use speaker wires that match your wall color. Other than that, hiding wires is overrated, in my experience. Wires disappear after two days - you'll not notice them much.
With equipment behind the viewers, you might also want to add an rf relay for your remotes, so you won't have to point them back over your shoulder to change channels. I got mine for about $50 at a local electronics store.
Lastly, I recommend you not ask the owner of the apartment first. Just fill your own holes before you leave. They will probably have to repaint anyway, so matching the paint won't be nearly as important as leaving them a smooth surface to paint on.