Thank you for all the detailed replies.
1) The deed is prepared by the seller's attorney who will also preside at closing.
2) Wise or not wise to close without an attorney is a separate issue, but from your answers it is legal. Thanks.
3) There is no third party involved, only the seller, seller's attorney, and the buyer (myself).
4) Unfortunately, I have no leverage to insist on warranty deed. I can politely request, but I probably will not get it. In our market quitclaim deeds are common. I bought my first house with a quitclaim deed 6 years ago. Also, it is such a strong seller's market (at least in my area) that I was competing against multiple offers, and the seller's position was to spend the least amount of money and do as little as possible. Is there any law that would require seller to produce warranty deed?
5) You didn't answer about MLC. You are saying it is "routine", but is it required by law? Yes, it protects me against possible liens from town, water department, unpaid sewer, etc. But this is already written in the P&S. "SELLER states that to the SELLER’s knowledge there are no lawsuits currently pending or threatened which will in any way affect title to the premises".
6) A Closing Disclosure not only provides information about a loan (which I don't have), but also shows, for example, taxes and other government fees (page 2 part E), which have recording fees, and transfer tax. In the previous closing we used HUD-1, which had separate lines for obtaining MLC, recording MLC, prepaid property taxes, and prepaid water minimum. I don't see these lines in the closing disclosure. Where does this information go now? Should we still use HUD-1? Also, there will be leftover in the heating oil tank. Where does this information go?
7) Paid taxes are public records on every property (at least in my county). Figuring out my share of post-closing taxes should not be difficult, because the amounts seller has paid is public information. To figure out prepaid water minimum I would need last water bill from the seller.
8) There is no mortgage closing. The only thing we did from the buyer's side in the previous closing (6 years ago) was the HUD-1, MLC, and the money, of course. MLC and quitclaim deed were recorded at the registry. I had an attorney who wrote HUD-1 and obtained and recorded MLC for me. I am trying to do it myself now.

