@Alison, Yes that is the birth details for Edith - I have a copy of the birth certificate, as well as copies for the siblings of Edith.
Eliza Victoria Crook is the mother of Robert James Crook, she died in 1938.
The Qld Edith May Crook is a different person - it seems to be a popular name!
@wivenhoe, Yes, the marriage in Sweden sounds very suspect. The death certificates for Robert James Crook and his second wife both list Sweden as the place of their marriage. I have always thought Sweden was used as a ruse to hide a marriage that didn't actually occur.
Re the WW1 record, the Redfern address represents the NOK address at the time of enlistment, rather than current address in Aug 1940.
@JM, The other death certificate is actually for Edith May CROOKS, although it appears as CROOK on NSW BDM. She is buried at Botany Cemetery, NSW (aka Eastern Suburbs Memorial Park). This Edith May Crooks never married and was 34 at the time of her death.