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 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.
Is it permissible to offer an absentee funeral prayer?
Yes, it is permissible to offer an absentee funeral prayer.
Is it permissible for the heirs of the person who caused the accident (The Killer) to fast (The intended as an expiation for unintentional killing) on his behalf if the latter died in the crash?
All perfect praise be to Allah the Lord of the Worlds. May His peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
One who dies while liable for obligatory fast, his/her guardians have the choice to either fast on his/her behalf or feed one poor Muslim for each day of missed fast. This applies to Ramadan and other obligatory fast, and since expiatory fast is obligatory, feeding a poor person for each missed day or fasting on behalf of the deceased is obligatory as well. And Allah The Almighty Knows Best.