Yes. A spouse admitted in L-2 status is employment-authorized incident to that status and may take a job without waiting for a separate work permit, although many employers still ask for a document to record. Unmarried children under 21 are admitted as dependants and may attend school, but they are not authorized to work.
The spouse's rights follow the principal's category, and so do the deadlines
This is one of the meaningful practical differences between categories, and it often decides which route a household prefers. Because the authorization arises from the status itself, a spouse can accept employment on arrival rather than waiting out an application; obtaining a document that evidences the authorization simply makes hiring easier in practice. Two dates deserve a calendar entry.
The first is any child's twenty-first birthday, since dependant eligibility ends there and a young adult who intends to remain will need a route of their own, most often as a student. The second is the principal's cumulative limit in this classification, seven years for a manager or executive, because when that expires the whole household's status expires with it. Families who discover both dates in the final year have far fewer options than those who mapped them at the start.