Is it permissible to openly announce the engagement of a woman who is in her waiting period (iddah)?
It is not permissible to openly announce the engagement of a woman in her waiting period (Iddah), but it is permissible to hint (imply) at it if she is in her waiting period after the death of her husband. And Allah Knows Best.
Is vomit among the nullifiers of Wudu (ablution)?
Vomit does not nullify Wudu, but it is a Najaasah (impurity) that requires rinsing the mouth and washing whatever became dirty by it since the prayer becomes valid only after the removal of Najaasah. And Allah Knows Best.
Is it permissible to pay the Zakah to my grandmother who lives alone in a rented accommodation with bad conditions noting she had three sons.
Zakah cannot be given to ascendants (parents and grandparents). It is obligatory for her sons and grandsons to provide for her, and they would be sinful if they fail to do so. And Allah Knows Best.
Is it permissible for the woman who is observing Iddah after the death of her husband to exchange calls with her relatives and husband`s family call?
All perfect praise be to Allah the Lord of the Worlds. May His peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
It is permissible for the woman who is observing Iddah* after death of husband to exchange calls with them; however, when she speaks to non-Mahrams , she shouldn`t be soft of speech and she should be straight to the point. This because Almighty Allah Says (What means): "O ye wives of the Prophet! Ye are not like any other women. If ye keep your duty (to Allah), then be not soft of speech, lest he in whose heart is a disease aspire (to you), but utter customary speech." [Al-Ahzaab/32]. And Allah The Almighty Knows Best.
[1] The iddah is a waiting period that a Muslim woman observes after the death of her husband or after a divorce. The Quran says: For those men who die amongst you and leave behind wives, they (the wives) must confine themselves (spend iddah) for four months and ten days.