Is it permissible for a person who hasn`t made ablution to call for prayer (Adhan)?
All perfect praise be to Allah, The Lord of The Worlds. It is disliked for a person with a minor impurity to call for prayer, but if he does , then it`s valid and sufficient. And Allah Knows Best.
Is it permissible for a woman to swear a false oath by the Holy Quran to please her distrustful husband?
It is forbidden for a person to do so since this is considered a false oath (Yamaan Ghamoos) which dips (Yaghmos) its taker in Hellfire if he/she doesn`t seek repentance, and he/she has to expiate for that oath. You should avoid any doubtful acts so that your husband`s chest remains clear from the evil suggestions of the devil.
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 the marriage contract considered valid if concluded at home by the marriage official (Ma`zon)?
Yes, it is valid as long as it is registered at the court.