When determining the result of an end may the thirds move a bowl out of the head before the total number of shots is agreed?
Yes. As the shots are agreed each shot bowl can be removed from the head. Law 22.3
The marker in a singles game has been asked by one of the players to measure all shots during the game. At the conclusion of one end, before the players have reached the head, the marker has recorded the score on the score card and kicked the bowls out of the head.
Is he correct?
No. No bowl must be moved until both players have agreed to each shot. Laws 22.2 & 42.2.13