Is it permissible for a person to give the Zakah (obligatory charity) to his daughter-in-law?
Yes, it is permissible for a person to give the Zakah to his daughter-in-law if she was poor, and none provided for particularly by whom are obliged to provide for her. And Allah Knows Best.
Is prayer mandatory on a dying person who goes through a state of unconsciousness, then wakes up?
If he wakes up and was conscious, then he should pray the way/manner he could, but if he wasn't, then he is exempted from obligatory prayer as Allah Says in this regard (What means): "On no soul doth God Place a burden greater than it can bear." [Al-Baqarah/286].
Does fasting on behalf of a deceased person permissible?
Fasting on belhaf of a deceased person is permissible, since the Prophet (PBUH) said: "Whoever dies while he still has some fasts to make up (of the days of Ramadan), then his heir (any of them) should fast on his behalf." [Agreed upon]. The previous answer is for making up missed obligatory fasts on behalf of the deceased. But if the fasting on behlaf of the deceased was for performing a voluntary acts of devotion such as fasting....is permissible as adopted by the majority of Muslim scholars and based on the above hadith as they stated "Every good dead intended to be on behalf of the deceased its reward will reach the latter." And Allah Knows Best.
Is it permissible for the mother to offer an Aqeeqah (the sheep slaughtered on the seventh day from the child`s birth)for her son?
Aqeeqah is due on the one obliged to provide for the newborn, and it is permissible for the grandfather, or the mother to offer the Aqeeqah.