How do install a suspended ceiling to an arched shaped room and how do you dry line a curved surface

To fit this grid ceiling your going to have to 'square-up' this arched/curved ceiling. At one end of the room set up a right-angled laser level which allows you to transfer the 90 degree angle from the floor to the ceiling. Next, mark the height of the new ceiling on the existing curved ceiling and set a laser level to that mark, this will allow you to horizontally laser level the entire room. Now, you are going to build a timber-framed structure set off those laser lines, essentially you are building a 4-sided squared structure and within it sits the new grid ceiling. The last problem is where the perimeter of the squared structure meets the curved ceiling but since this is in the horizontal 'flat' plane therefore no special bending of the plasterboard is require so all that is needed is to scribed one end of the plasterboard where it meets the curved ceiling and the other end of the plasterboard is fixed to the new squared timber structure. I hope that i have understood your job correctly and what i have said is not confusing and makes sense. All the best.

Get somebody who knows what they are doing

Depending on the type of suspended ceiling it won't make any difference of the existing ceiling as you'll put your surrounds in level then infill and bring down your supports to that level.