I was on my school's FSAE team (FSAE is the competition you were talking about) since it's inception 4 years ago. I've since graduated.
I think you'd be dreaming to build it for anywhere near $1000. Unless you took some serious short cuts (read: dangerous). Our first car, which was pretty crappy cause it was cheap and had a lack of seriously involved engineering design (cause getting it built in time was tough enough), was still over $10000.00. Now, seeing as you don't have to follow the rules and regulations that we had to, some things could be eliminated. But you'd still be looking at over $3000.00 for a VERY basic setup.
We used a Suzuki GSXR600 motorcycle engine.
You'd definately want to stay away from Aluminum for more than just the TIG reason. Aluminum is too weak for this application without some serious engineering. I'd stick with mild steel as it's probably the cheapest, can be MIG'd, and is just as easy to work with.
Working with a small car's steering/suspension would be WAY too heavy, but you could use a small car's hubs and maybe even uprights. Most other suspension components would have to be fabricated. You can buy a quality small steering column for pretty cheap online (this is what most FSAE teams do).
I'd definately get involved with your school's FSAE team as it's a very rewarding program and there's tons of knowlegde to learn. It's probably my favorite memory of university.
Just a little list of some things off the top of my head.
-Buy an engine with the stock wiring harness and ECU.
-you'll need a motorcycle muffler (you can just buy a cheapie)
-fab an exhaust (header)
-fan an intake (header)
-you can use the throttle body from the motorcycle your engine's from
-buy or fab a steering wheel
-fab a steering column (will most likely need to get the end splined for the steering rack)
-buy a steering rack
-buy or fab uprights and hubs
-buy a wheel and tire package (look into Hoosier) (I wouldn't go larger than a 13" rim)
-you'll need a chain and spindle (you're looking at a min of $500.00 for a differential, so the cheapest and crapiest route would be to go without - shudder)
-if you go with a differential, you'll need splined and hardened driveshafts
-you'll have to buy brakes (at least the calipers and pads) rotors can be made (ie. you design the rotor and get it laser cut - simpler than it sounds)
-buy bearings (make sure they fit in your uprights)
-make a jig to hold your chassis together for welding
-buy chassis tubes and cut and weld to your design
-make sure the chassis has roll bars (matter of fact, look at the rules for the FSAE challenge and copy some of the chassis safety from it)
This is just off the top of my head. I hope it helps with your research.
