First, handle drainage. If you want to install a sump pump, perimeter drain, or water proof the walls, now is the time to do that. You should also install any plumbing drain lines at this point.
Next, framing. Concrete transmits moisture, so use pressure treated, and a styrofoam underlay that would normally go under the sill plate, to keep the walls dry. With pressure treated wood, you need galvanized nails to avoid a chemical reaction that would eat a normal nail. Make sure the framing is designed for the drywall (16" OC studs, nailing edge in the corners and around the ceiling perimeter). And make sure the walls are positioned/sized for any plumbing or other utility lines. For exterior walls, the thicker the wall (2x6 instead of 2x4) the more insulation you can include.
At this point, utilities go in, including plumbing, electrical, hvac, and communications. I'm a strong believer in conduit for running communications lines (like ethernet, catv, and phone) so that you can upgrade those without opening the walls or running wires all over the floor. Note that with more outlets, you will need more circuits, and copper isn't cheap.
Insulation is done after all the utilities are finished. If you have water proofed the walls, then I don't have much against the standard fiberglass insulation. Actually, I like fiberglass because it's easy to work with as a DIYer, relatively cheap, and replacing a piece or temporarily moving it (e.g. in the attic) is simple. However, with moisture concerns, or if you live in a northern climate with harsh winters, then I'd favor the spray foam. The spray foam has the downside of being difficult (impossible?) to remove, but it functions as a vapor barrier and leaves no cracks that air can get around.
After this, you have drywall, paint, doors and trim, and touch-up paint. I like to paint the walls and trim separately so there's less work on edging.
As for the flooring, you would usually do this before or after the trim. With hardwood, you would often do the flooring first, and then the trim goes up tight against it. With carpeting, you would install your trim 1/2" above the floor and the carpet installers would install the carpet right under that.
I lived in a house that had standing water once or twice a year. It had a pad with berber(?) carpet on top. It dried easy with a space heater and fan within a day of these wet periods. Usually a 5x10 foot area got wet. When I went to pull up the carpet both the carpet and pad were in perfect condition. I asked a friend who had done installations for years and he said that I had an open-celled padding with no moisture barrier and if they are dried out thoroughly within a reasonable time he has never seen mold.
So new house I am in is basically about the same as yours. I am putting down carpet in part of the basement with pad. Almost all pads are open-celled and you would have to ask for one with a moisture barrier - so if you go to a carpet store and pick out their normal padding that is the right thing.
Also I am using Trafficmaster Allure Ultra on part of the basement which I have used on jobs. I am not a HD homer - but I have used this stuff and it is awesome. It is pure rubber and even if you got mold (you shouldn't) it would just clean right off.
As for your $1000 budget. You are going to have to get some very low grade carpet to meet that. Do not get non-floating vinyl. If you have to install the vinyl to the floor and you do have moisture problems you are in for a big mess. The recommendation I have above is $1800. Also a great website for flooring builddirect.com has other all rubber options but they may make you have a minimum order.
Best Answer
Sounds like a hack-job approach to me.
That wall could be built from concrete blocks (aka CMUs or concrete masonry units), with bond-block rows and reinforcing steel, and grouted full, making it effectively solid, without abandoning forms in place - or rather, the forms would be the block wall.
Of course, poured concrete contractors and masons are not usually the same people, so this would not be a solution a concrete contractor would be liable to think of.
Alternatively, a form never intended to be removed and not made of wood that will rot - the insulated concrete form or ICF. Two sheets of styrofoam tied together.
But your pictures imply that this is a done deal.
If termites are a problem in your area, they may find that form tasty.