You can do it however you want - but there is always a cost and consequence. Maybe I can help put a finger on some of the costs and consequences.
1 - A fully enclosed room has several advantages. If defines an acoustic space in which you can mold the sound. Much (most?) of what you will listen to in the theater will be sound reflected from the walls and ceiling of the room. Your brain is very sensitive to the nature of the reflected sound. For the your brain to work with the sound encoded on the track to recreate the "space" of the soundtrack, symmetry and consistency are important. If there is no wall in the rear, your brain will know it, and the sounds from behind you will never have the same character as others. It's hard to put a finger on, and you can enjoy the soundtrack and experience in either case, but how about this: this is one of the few things that BIGmouthinDC has said he would change about his theater if you were to rebuild it.
The second major advantage to an enclosed space is the possibilities for sound isolation. A larger open space invites windows, appliances, and foot traffic - all of which bring in unwanted sound (and heat) and let out nuisance sound to other areas. Your requirements for this are your own, but overcoming these challenges is one of the most significant design/engineering hurdles in dedicate theater building. Be realistic about your uses and needs (as well as your neighbors' and the rest of your household's).
2 - Dark grey for the drywall surfaces is a pretty good choice. Look at LRV (light reflectance value) for the colors/paints you are choosing (I have only seen this bit of data from Sherwin Williams, but it's probably available for other makes - you own't find it at Lowe's or from Martha Stewart. A lower LRV means less of the light from your projector that is reflected to the walls from the screen will reflect back onto the screen and wash out the dark areas of the image. Preserving intrascene contrast is a difficult challenge, but that process can also be aided by certain types of screen technologies. The carpet falls into the same category, obviously, but has some simpler solutions. You could cover it with an area rug near the screen. In fact, this philosophy can be carried to the walls and ceiling, with paint choices.
3 - The cheap ways of dealing with theater acoustics are mostly limited to absorption, and do-it-yourself. There are lots of resources around here to help you visualize the possibilities. As a rule of thumb, try to allocate 4 inches of thickness for wall treatments - keep that in mind when you lay out the seating and other features of the room, like wall sconces or columns or wainscot.
4 - I'm no good for decor ideas, so I'll leave that. You should check out all the "show me" threads in the stickies.