Thank you for the compliment :) we try, I am sure some agents will kick in soon and offer their services for when you are ready. In the meantime, I'll do my best to explain some things about the market here. On Trulia you can see recently sold properties by checking off "recently sold homes" on the left rail. The thing to note when you see those homes is that the typical closing in NYC takes 60-90 days due to title searches, and the red tape of "law". Therefore, the prices you see - even if they closed last week - were prices that were accepted in november to december of last year. So, take that knowledge, compare it to the homes available in the market now - make sure that property type, bedrooms, zoning etc are similar.
In the end, buying a place here can be a bit difficult to navigate due to the intricacies of NYC real estate. I highly recommend enlisting an agent in your search early to help you gauge the market, you may want to set-up some trulia alerts as well - if you have not done so already. We do have "buyers" agency in NY, it is not req'd but at times recommended - there are a few threads about buyers agency here on Voices.
In lieu of cash, and being a fast closing - it depends on the seller motivation. If they need to close in 30 days or less, or if you get lucky and find a distressed home seller it will make a huge difference. It is not likely however that you would be able to find out, but you can tell your agent that you can close in 30 days and that may help with getting a price reduction - especially if the home is on the market for more than 6 months.
Finally, with all of the above, you can figure out how low you should go. Placing a % on it is not my idea of good advice. Some homes are priced perfectly so full asking or above asking could be right, while others are overpriced, and 20% off would make sense. I recommend that when you get here, or even before you do - start looking at virtual tours - video, photos, etc and educate yourself as much as possible, so you will know what 800K SHOULD get you and what 650K should get you.
I personally saw about 30 homes over a year ago when I bought my current property, and I still feel like I got a great deal and my home actually appreciated in value - compared to what I paid for it.
Hope this helps!