Is it permissible for a Muslim to slaughter an Aqeeqah on behalf of someone else, and offer it to him as a gift?
In principle, the guardian is the one who should offer the Aqeeqah (the sheep slaughtered on the seventh day from the child`s birth) because he is obliged to provide for the newborn, and it is impermissible for anyone else to slaughter it on his behalf unless with his consent. However, it is permissible for a person to offer the sheep, or its price as a gift to the guardian of the newborn, and then the latter can slaughter it, or deputies someone else to do that on his behalf.
Who are the blood-relatives with whom kinship ties should be maintained?
Blood-relatives are those from the side of one`s father and mother: grandfathers, grandmothers, uncles, and aunts. And Allah Knows Best.
Is it permissible to perform Tayammum for a month due to a medical condition that requires avoiding water?
Tayammum is permissible so long as the above excuse lasts. And Allah Knows Best.
Is it permissible for the young woman to turn the suitor down if she had no feelings for him?
Yes, it is permissible for her to do so.