What challenge does he give the Court? Why does Gawain interrupt King Arthur? What agreement did he make with the Green Knight?

The Green Knight challenges someone in the court to play a game, and whoever agrees will be allowed to strike the Green Knight on the spot, in the middle of the court. In exchange, the Green Knight will strike a return blow upon the volunteer a year and a day hence.

Because no one else with play..... Arthur agrees to the challenge.

Gawain onjects to Arthur putting himself in danger, and he accepts the challenge on Arthur's behalf.