Christ exercised mercy by healing the sick, and thus saving them from their dreadful diseases. His mercy cured many cases of blindness (Mt. 9:27-30) and by extension all the other illnesses He healed. He delivered people from demon-possession (Mt. 21:21-28, 17:14-18, Mk. 5:19), and God the Father in mercy healed a woman of infertility (Lk. 1:58). In contrast, unbelief toward Christ prevented Him from extending His healing mercies, in at least one scenario (Mt. 13:57-58).
The Lord shows mercy, and gives His kind gifts, by His own free, unfettered choice, and according to His own principles. He can never owe men or nations His kindness because of their good works (Ro. 9:14-18), a principle illustrated by God’s choice of the nation of Isaac over the nation of Ishmael (9:6-8) and the ordination of the line of unborn Jacob over the line of unborn Esau (9:10-13, Mal. 1:1-3). God revealed the backside of His passing glory to Moses (Ex. 33), not because Moses earned it, but as a marvelous favor.
God shows mercy to the repentant, regardless of who they are, and hardens the unrepentant, regardless of who they are (Ro. 9:18, see Peter’s words in Acts 10:34-35). Peter realized God's mercy-principles at work when he met the Roman centurion Cornelius. God sent the Gospel to Cornelius because Cornelius was devout, God-fearing, and God received his alms (Acts 10:1-5). In other words, Cornelius was seeking the Lord diligently, and God did not let him down (as He promised Israel in Jeremiah 29:12-13).
Does God give saving mercy to non-Jews on different grounds from how He gives it to Jews? Does God show pure saving mercy to Gentiles, but then require a mixture of mercy and good works from the Jews? No, Paul says God combined (aorist tense, mingled, mixed) Gentiles and Jews together in an equal state of disobedience, so that He might show mercy to them all (Rom. 11:30-32).
God’s saving motive toward us all was rich mercy and great love (Eph. 2:4), enacted through the death, resurrection, and ascension of Christ. Salvation is entirely by God’s mercy, not at all by legal performance (Titus 3:5). The new birth is the experiential expression of God's mercy in the soul (1 Pe. 1:3).