What is the ruling on someone who prays Fajr late, after sunrise, knowing that they wake up on time but return to sleep and do not perform it within its time?
It is obligatory to perform the prayer within its prescribed time. If someone wakes up and knows how to act, they must perform it before the time ends. If they delay it, they are sinful and must make it up. And Allah Knows Best.
Is a menstruating woman obliged to pray during menses, and should she make up missed prayers?
All perfect praise be to Allah, The Lord of The Worlds A menstruating woman is prohibited from prayer during her menses, and she isn`t obliged to make up missed prayers. 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.
Does ill-gotten money hinder the answering of the Du`a (supplication) and the means of subsistence?
Sins hinder the answering of the Du`a, and deprive the sinner from means of subsistence; whereas, acts of obedience facilitate making a living. Allah, The Almighty, Says (What means) : "And for those who fear God, He (ever) prepares a way out. And He provides for him from (sources) he never could imagine." [At-Talaq/2-3].