Sounds easy - but do I need a registration / plate for the trailer ?
Answer: In some states, a license plate is required for the trailer. Depending on which ones you plan to travel through, you may want to proceed with getting a license plate anyway. What you will not be able to get is certification from the manufacturer on trailer pulling as others have described above.
However, if you are working with a dealer in the US, that would be my first contact for obtaining the title/bill of sale and state licence plate.
If yes - how I could manage this, can it be registered by the dealer before delivery ?
Answer: See the explanation above.
I have a contact address in Ohio, could be useful for registration, but of course I need the trailer in Halifax to load up my luggage.
What happens when I leave USA downwards to South America ?
Answer: Your contact in Ohio would be a good start for register and, possibly, transporting the trailer to Halifax. As for south of the border, whatever license plate you put on the trailer and have on the bike should suffice.
One thing you have not mentioned and may not have considered is insurance on bike and/or trailer. For the US portion of your trip, there may not be that much difference in what you have at home and what may be required in the US. However, that will change once you cross into Mexico. Others have traveled there and can speak to recent efforts but when I attempted to go down some years ago, my insurance company would not cover the bike, our trailer or us in Mexico. We changed our trip plans and headed North.
Again, you may have looked into this but if you haven't, it's something to consider.
Hope you have a safe, happy and sight-seeing filled adventure in your visit and keep it between the ditches.