1) if he directly informs US about my moving out, how soon will I get letter from US telling me to leave dk?
That can take a while. They know that people surprisingly often reconcile.
2) if he doesn't contact US, but fills "petition for legal separation" in my new address, how soon will I get such letter from US?
You will not get the letter until you have actually divorced (has been practice until now, at least).
3) once divorce is done, how soon will I get such letter from US? how long will be the grace period to stay in dk legally before the deadline of my leaving?
It has been known to take several months, but mostly in cases where there are children.
4) if I plan to remarry in dk. for example I receive such a letter from US on 1st Feb telling me to leave dk no later than 28 Feb. Am I allowed to register new marriage only BEFORE 1st Feb? what kind of special document is needed for marriage registration during the grace period?
You of course cannot re-marry until you are divorced. You can apply as soon as the ink on the divorce paper is dry, and probably get married within a couple of weeks. There is no special document for your situation. What you need is your ID and the divorce paper.
5) If I appeal against US' decision of revoking my residence permit, can I stay in dk and work legally while waiting for the final decision? Will I be covered by the public health care system?
On what grounds would you appeal? If you have no legally valid reason to believe that the decision is wrong, an appeal will be rejected swiftly and mercilessly.
6) Are all above decisions from authorities send by post mail or e-bok?
Probably both.
7) what is the risk for him for not reporting my moving out?
Probably not great. I have not heard anything about huge fines.
BUT - You say you have ‘your office address’ at your hitherto home. If you have a business of your own, you could perhaps use this:
https://www.nyidanmark.dk/en-GB/You-wan ... retainment