1. On what basis does God bless mankind?

Simply according to His good will, for He makes it rain and causes His sun to rise on the good and the evil (Matthew 5:44-45)
According to His good will, however, He is attentive to the prayer of those who seek His righteousness (1 Peter 3:10-12)

2. Of these parents miraculously blessed by God with a child, which ones certainly prayed before receiving this blessing? (We recommend reading the biblical references at the end)

Abraham and Sarah (Genesis 21:1-4)
Isaac and Rebekah (Genesis 25:20-21)
Jacob and Rachel (Genesis 30:22-24)
Manoah and his wife (Judges 13)
Elkanah and Hannah (1 Samuel 1)
Zechariah and Elizabeth (Luke 1:13)

3. The characters of the "New Covenant" below have been richly blessed by the Lord; which ones are described as having performed works that speak in their favor? (We recommend reading the biblical references at the end)

Tabitha, who was full of good works and almsdeeds which she did (Acts 9:36, 39)
Cornelius, the centurion, offering prayers and alms to God (Acts 10:1-6)