Is it permissible to propose to a pregnant divorcee who is observing her iddah (waiting period), and is it permissible to conduct only the administrative marriage contract during her iddah to facilitate some administrative matters?

If "leaving" refers to a man proposing to a woman, then a woman observing iddah (waiting period) is not permitted to leave her house, because she is commanded to remain in it until her iddah ends. She is allowed to leave for her own specific needs, not for the needs of others.

As for proposing to a pregnant woman observing iddah: if the iddah is revocable (iddah rajiyyah), it is not permissible to propose to her, neither explicitly nor by insinuation, by consensus. However, if the iddah is irrevocable (iddah ba'in), then insinuation is permissible, but not explicit proposal.

Regarding a pregnant woman observing iddah entering into an administrative contract: if the intent is a marriage contract, it is strictly forbidden. However, if it is a contract of sale or lease, there is no objection if it is for her own specific needs.