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.
What is the ruling on buying a car from someone for less than the market price, because they are in a hurry to travel and want to sell their belongings at a lower price due to their urgency?
Yes it is permissible; however, a Muslim shouldn't take advantage of other's difficult situations.
Is it permissible for us to sever ties of kinship if our blood-relatives` gatherings involve acts of sin?
Maintaining kinship ties is mandatory, thus if visiting your blood-relatives hinders their sinful acts , then you should do so. However, if their sinful acts persist while you are at their gathering, then it is sufficient that you maintain kinship ties via telephone and the like. And Allah Knows
Is it permissible for a man to propose to a woman during her `Iddah (waiting period)?
All perfect Praise be to Allah, The Lord of The Worlds, and may His Peace and Blessings be upon our Prophet Muhammad and upon all of his family and companions.
It is impermissible to propose to a woman in her `Iddah, but there is no harm in making an implication to such an end in case she was observing it due to the death of her husband. And Allah Knows Best.