Is it permissible for the big brother to force his sisters to wear Hijjab (Islamic wear) if their father didn`t do his part in this regard ?
All Perfect Praise is due to Allah, The Lord of The Worlds
The father`s attention should be drawn gently and politely, and the sisters should also be reminded with good words since Allah, The Almighty, Says (What means): "O ye who believe! Save yourselves and your families from a Fire whose fuel is Men and Stones." [Al-Tahreem/6]. And Allah Knows Best.
Has Islam made it incumbent on us to give our children particular names?
No, it hasn`t, but it is desirable that we give them good names.
Is hair extension permissible?
All Perfect Praise is due to Allah, The Lord of The Worlds
It is impermissible for the man and the woman to apply a human`s hair as extension, but using artificial hair is permissible for the married woman after obtaining the approval of her husband so long as non-Mahrams don`t see her. And Allah Knows Best.
Is it permissible for a woman in her waiting period (Iddah) after her husband's death to leave her house and mix with non-Mahrams?
It is not permissible for a woman in her waiting period (Iddah) after the death of her husband to leave her house except for a necessity, such as for medical treatment, nor is she allowed to mix with non-Mahrams. A woman is not permitted to mix with anyone other than her husband or Mahrams, whether she is in her waiting period or not. However, there is no harm in accepting condolences from those who visit and responding to their inquiries about her well-being. And Allah Knows Best.