You application won't be reviewed until all your materials are in - including your LSAT score. You're not 'complete' with out that LSAT score, so submitting the rest of your materials earlier doesn't help you. Since you're taking the September LSAT, the earliest you can go complete is late October. Columbia's ED deadline is Nov 15th I think, so you should be fine.
It's a good idea to ask for LORs now - you'll need them regardless of your LSAT score, and it's important to give your professors enough time to write them. I'd probably go ahead and request your transcript too - it will probably take all of fifteen minutes, might as well get it out of the way. I wouldn't worry too much about the application forms right now, since your list of schools isn't finalized. As for your PS, it doesn't hurt to think about it - but your number one priority at the moment is making sure you do well on the LSAT.
I'm of the opinion that there is very little benefit to applying early decision. Since law school admissions are generally rolling, you can apply regular decision early in the cycle (before thanksgiving) and have your application viewed before most others. Since HYS don't have ED, many people who apply ED don't do so to their number one choice. This seems like a bad idea to me. If Columbia, even at full price (they may give you money, but they have no incentive to do so since you've committed to going already) is your number one choice beyond anything else, then yeah, go for it. I think ED programs are mostly for the school's convenience though - they're able to secure applicants who are 100% going to their school early on - rather than yours.