Yes. An L-2 spouse is employment-authorized incident to status, so employment is permitted without a separate work permit, and evidence of that status is what an employer verifies. Children under 21 hold derivative status that allows study but not work.
The category the principal holds decides what the spouse may do
This is one of the sharpest differences between routes, and it often decides which one a household prefers. Where the principal transfers within a company, the spouse's ability to work follows automatically from status rather than from a separate application, which matters a great deal to a family moving on the strength of one salary. Practical friction remains.
Some American employers and payroll systems are unfamiliar with employment authorization that arises incident to status, so the spouse should expect to explain the basis and may still want the document that makes verification easy. Children are derivatives only while unmarried and under 21, and a child who reaches that age during the assignment loses derivative status and needs an independent basis to remain, which is worth mapping against a seven-year ceiling rather than against the first petition. Each traveller's own intended activity, not the household's convenience, determines what is permitted.