Yes. A spouse in E-2 status is employment-authorized incident to status, so the authorization arrives with the classification and is not restricted to the family enterprise. Children under twenty-one may accompany the family and attend school, but are not authorized to work.
One household, three different sets of permissions
The investor's own activity is defined by the business she is developing and directing, and she should not be taking outside employment that is inconsistent with that. Her spouse is in a materially different position and may work for any employer, which for many families is what makes the route viable while the enterprise is still building. Employers verifying eligibility will want to see the admission record, so it is sensible to keep the marriage certificate and status documents accessible.
The children's position is the one that needs a diary entry rather than a document. Derivative status ends at twenty-one, and because E-2 can be renewed indefinitely, a family may be in the classification long enough for a young child to reach that age inside it. Planning a student's own route ahead of that birthday avoids an abrupt loss of status.