yummm...$12k cash to spent on building an integra. I wish I had more free time to help you find a car and parts, i'm always up for a good challenge. Honda-tech.com is a good source for parts, just make sure the seller is reputable, ask for references, etc.
First of all, each step you take is probably going to cost about 25-50% more than you originally expected.
Look for a clean titled '94-95 RS/LS. That will probably be your cheapest option (base cost & insurance) and if you want the newer frontend the '97-'01 bumper and headlights will bolt right up w/o a problem. Expect to pay $3-5k, depending on the mileage and condition. You could always look for a stolen recovery from a auction for normally much less.
I'd start with getting it rolling, an ITR 5 lug conversion is something you can do for $400-800 or so which will give you alot more options on "nicer" wheels, better suspension setup, and enable you to sell the original brakes to a civic owner. Tons of suspension is available for either way you go....I'm a big fan of Tein but by browsing the forums you can just find a great deal on something out there. I'd set aside $1k for a basic coil-over dampener setup and installation, whatever the brand may be. Of course you can spend upwards of $1k on each corner. If you wanted to keep the original suspension I can get you a full ground control coil-over dampener setup (not the sleeve) for about $800, which is normally over $2,000 then add another $500-1000 for a decent brake upgrade. I'd say a minimum $2k for wheels, brakes, and suspension...if you look really hard.
Engine...
K20a2 from the Civic SI will be alot cheaper than the K20a3 from the RSX but with either one you'll need a mounting kit and several other things that aren't already on the DC2 body you buy. It will probably be $4-5k+ for the swap (and transmission & parts) then add in $1k for labor and unexpected expenses. I've seen clean (not stolen) A3's with tranny sell for under $4k several times.
I like the B-series because one day (!!!) the k-series will eventually take over the market and the price of b-series parts and upgrades will drop. Unless you're going turbo i'd probably refrain from the [GS-R] B18C1. For the cost of GS-R or ITR swap you could build a great LS/VTEC or B20/vtec turbo. You'll have alot more expenses down the road with a turbo setup though.
With a mid-grade suspension and decent engine setup you've practically spent your budget but could easily spend double. A full repaint done correctly will cost about $3k...it can be done under $2k but you get what you pay for. Fix up the interior, maybe a rollcage, etc...it all depends on what the main purpose of the car is...show, go, or a little of both.
I spent your budget on just my car, '01 GS-R with 37k miles then put in a brand new B18C5 rolling on Tein Flex...let's not add my stuff up $$$
