May a woman in her ʿiddah receive a proposal or become engaged?

A woman in the ʿiddah of a revocable talaq (raj'i) may not be proposed to at all; a woman in the ʿiddah of an irrevocable talaq (ba'in) or of death may be proposed to by hint only, not by an explicit proposal.
Allah permits hinting (taʿridh) to a woman in her ʿiddah, but forbids making secret promises and forbids the contract before the ʿiddah ends. A marriage contract during the ʿiddah is void.
A woman in a raj'i ʿiddah is still another man's wife and may be taken back, so even hinting is forbidden. Only after the ʿiddah ends may a formal proposal and engagement take place. See ʿiddah & rujuk.