Do cadets participate in operational missions? If so, what type of missions? Under what circumstances would a cadet be sent on a mission?

Cadets often participate in missions. See guidance below from Section B, Paragraphs 1-10e of CAPR 60-3.

e. Use of qualified CAP cadets is encouraged as much as possible on appropriate missions. Cadets should be trained in the various functions of mission operations and support as permitted. Cadets qualify no differently than adult members in emergency services qualifications, and can be properly utilized in age-appropriate scenarios. Additional guidance for employing cadets on missions can be found on the CAP/DOS website at  Cadets in Emergency Services.