Yes to both. An L-2 spouse is employment authorized incident to status and may work for any employer, and accompanying unmarried children under twenty-one may attend school at any level. The children themselves hold no work authorization, and their derivative status ends when they turn twenty-one.
School enrolment is straightforward; the twenty-first birthday is not
Practical enrolment questions are usually resolved by the school district, which asks for proof of residence, immunization records and previous transcripts rather than immigration documents beyond status. The immigration questions worth planning are elsewhere. A spouse's authorization arrives with the status, which means it is not employer-specific and does not depend on the principal's company.
A child's position is more fragile, because derivative status ends at twenty-one regardless of where the school year sits. Where a child intends to continue at an American college after that point, the student route has to be arranged in advance, and the record-keeping requirements of the student system are handled by the institution rather than by the family. A five-year ceiling on the principal's status can also arrive mid-degree, so the two timelines should be drawn on the same page.