I'm in the midst of a similar project right now. In the case of the stem, my stem was steel. It was super heavy! I'm converting to a threadless headset. Only saves about 20 grams on the headset but it saves me almost 300 grams on the stem. My drop bars were also steel and a hefty 646 grams! The new aluminum ones are only 303 and they were only about $25 brand new. Converting it to threadless also means that I had to spring for a new fork. In your case if you want to keep the stock threaded setup, you need to look for some sort of aluminum or titanum stem. this should save you about 200 grams. Aluminum drop bars are cheap, prob no need to buy them used but it can't hurt, heck, if you have friends that are bikers, one of them probably has most of this stuff that they can just give you. worrying about brake pads at this point seems a little bit pointless, get the bike built and riding first than settle on those details.

As for your shifting, you won't be able to use the derailer that came on the bike if your going with modern brifters, you'll have to get some sort of newer derailer. I have limited experience with modern brifters but i can tell you that the times i've uesd Sora brifters I thought they totally sucked! If I were going to splurge somewhere, that would be it. With that said, brifters are EXPENSIVE, hopefully you can find some used. Does this bike already have a crank in it? If so, I wouldn't be replacing that until I had some major problem with it. Only other reason to replace an otherwise functioning crankset is you mention that you have some sort of injury, if that affects your cycling ability, you may want to go with a compact double crankset instead of a standard double, especially if you'll be riding hills. But I say save on the crank, spend on the brifters.

Does your $600 budget include the cost of powdercoating?

Regarding wheels, I'm 235lbs and haven't had any problems with my entry level wheels and i live in a city with tough streets, I think just about any rim with 32-36 spokes should be fine for you. I bought some entry level alex single wall rims from aebike.com and only paid 30 bucks for each of them. You say that the bike is light even with the steel wheels. My 83 nishiki was 32.0 lbs. I put the $30 aluminum wheels on the weight dropped to 28.4 so there is huge weight savings for cheap. got an sram chain from aebike.com as well for 13 bucks.

Do you have a kitchen scale or some sort of gram scale? weigh out your brakes and see what they weigh. I have old school diacompe brakes and levers and they are as light as just about any modern part so you may be all right with keeping existing brakes.

Biggest areas for you toget the bike riding but save money would be to go with friction downtube shifters and keep the existing derailers and if you have aluminum brakes like my diacompes just put them on a new aluminum bar.

BTW, show us some pics of what you got!!!