When Rabbit get shot he's wearing a brown patient smock. When his bullet-riddled body is shown on the ground he's wearing green scrubs.
When Trancer hunters are sent back in time, their minds are in the body of their ancestors. If the ancestor's body uses the chamber to go forward in time, it would create a paradox because if the ancestors has no children, the Trancer hunter would never be born and thus never go back in time.