Is it permissible for an engagement to take place without a legitimate marriage contract, and is it permissible for the fiancé to go out with his fiancée or visit her in her family's home without a mahram?

There is no objection to proposing to a woman without concluding the marriage contract, provided she is not in her 'iddah (waiting period). However, being alone with her is forbidden because she is still a non-mahram (stranger).