Check out the web page for the program. Lots of schools publish this information. You could also email the program and inquire. The biggest thing, which I'm sure you've heard, however, is fit. Some schools might only accept 5% of their applicants, but your interests might match up perfectly. Other schools might accept 35%, but you might not have matching interests. This is where the SoP comes in. That is your chance to prove to the school that you have done your research, and you will succeed there, not just because you are smart and hard working, but also because you make sense within their program.
My $.02 is that refining an existing writing sample is always time better spent than writing a brand new paper from scratch. You *definitely* don't need two writing samples, so focus all your energy on making the existing essay as strong as possible. I'm planning on digging back into mine, doing a bit more research, and generally slashing parts of it to bits.
This. LoRs are NOT held in as high regard if you don't release the recommendation. If you don't think a professor will give you a shining LoR, then find someone else. It is always a terrible idea to retain the rights to see your letter.