His status is derivative of his wife’s as investor, so it ends when the marriage ends. His employment authorization incident to status ends with it, and the children’s derivative status depends on the parent who holds the principal classification.
Derivative status lives and dies with the marriage
The spouse of a treaty investor is admitted in E-2 dependant status for the same period as the principal and is employment-authorized incident to that status, which is why the husband can cook in the venue’s kitchen, or work anywhere else, without a separate permit. That permission is a consequence of being the investor’s spouse. If the marriage ends, his E-2 dependant status ends and so does the work authorization, and he must leave, change status or qualify in his own right, for example by investing in an enterprise he owns and directs.
The children’s E-2 dependant status continues as long as one parent remains the qualifying principal and they are unmarried and under 21; they may study but not work. Ownership can be structured from the start so that each spouse holds a qualifying interest as a principal investor, though each must then develop and direct. The more common answer is to plan the long-term category early, since permanent residence, once granted, does not depend on the marriage in the same way.