Is it permissible to hold a small engagement party and prevent mixing of genders in it, and to inform friends about the engagement? And would they be sinful if they announced the engagement out of necessity due to the custom of the people of the country?

There is no objection to holding an engagement party and celebrating it, nor is there any harm in informing female friends about it. The narration regarding concealing the engagement is weak. As for men and women gathering in one hall with their separation and the presence of a barrier between them, there is no objection to it.