Having read the play, do you think Kate knew from the start Larry was dead? Why (not)?

I think Kate knows deep down that her son is dead. Her delusions about Larry's disappearance and her vehement self-denial are symptomatic of greater issues than just a grief-stricken mother's inability to cope with the loss of a child. Nervous and suspicious, Mother has taken on the burden of her husband's secret while he presents the face of an untroubled conscience to the world, while she suffers from headaches and nightmares. I think all of these stresses contribute to her denial of Larry's death though the truth is buried somewhere in her pain.