Was Paul's birth part of God's plan?

Yes, Paul's birth was part of God's preordained plan. He was on God's mind in eternity past, and at the appropriate time his mother conceived. God formed him in the womb and brought him to birth. This was God's purpose to reveal His Son in Paul.

Les explains that when Paul writes God separated him from his mother's womb, it means Saul of Tarsus was already on God's mind in eternity past. At the appropriate time, his mother conceived, and the baby in her womb was all part of God's preordained plan. God formed him in the womb and brought him to birth. He called Paul by His grace, not because Paul deserved or earned it. God's purpose was to reveal His Son in Paul, the resurrected, crucified, and risen Christ.

Galatians 1:15

But when it pleased God, who separated me from my mother’s womb, and called me by his grace,

Was Saul's conception and gestation part of God's plan?

Yes, Saul's conception and gestation were part of God's preordained plan. God had Saul on His mind in eternity past. At the appropriate time his mother conceived, and God formed him in the womb and brought him to birth, calling him by grace.