Is it permissible for us to sever ties of kinship if our blood-relatives` gatherings/meetings involve acts of sin?
Observing kinship ties is obligatory, and paying your blood-relatives a visit causing their sinful acts to be hindered , then you should do so. However, if their sinful acts continue while you are at their gathering and they didn't respond positively, then observing kinship ties via telephone and the like will suffice in this case. And Allah Knows
Is it permissible for a Muslim woman to wear trousers at home?
Wearing trousers at home isn't prohibited so long as non-Mahrams do not see her. And Allah Knows Best.
Who has the right to child custody when final divorce takes place?
The wife has the right to child custody until her children reach the age of puberty. However, if she gets married, the right of custody becomes that of her mother`s.
Is it permissible to make ablution without answering the call of nature after waking up?
Washing front and back openings (Istinjaa`) is not a condition for the validity of ablution because it is done to remove Najaasah (impurity), thus if there is no Najaasah, there is no need for that, and then it is permissible to make ablution without answering the call of nature. However, prayer is invalidated if there is Najaasah on the anus ,or the front opening, so it (Najaasah) must be removed for the prayer to become valid, and not the ablution.