18:18 (A) What is the vow referred to here?

Paul's vow here was a special pledge of separation and devotion to God.

The shaving of his head was based upon the Old Testament Nazarite's vow. Nazarites in the Old Testament separated themselves from others by total consecration to God with a special vow of penitence and devotion (CP Nu 6:1-5, 13-21).

Scriptures do not tell us why Paul made a vow, but most Bible commentators believe he did it in a spirit of thanksgiving for all that God had done among the Gentiles through him, and that the vow would be fulfilled when he returned to Jerusalem (CP Ac 18:19-21).

See also author's study Paul the Apostle - a Chosen Vessel unto God in his book Advanced Studies in the Christian Faith (Volume 2).

(B) By being named first here does that mean as some claim that Priscilla was more prominent in the early church than Aquila?

No, but those who argue for women leaders in the church curiously interpret the fact that three times in scripture Priscilla's name is used before her husband, Aquila, as proof that she was more prominent in the first century church than he was (CP Ac 18:18; Ro 16:3-4; 2 Ti 4:19).

It is hard to understand the logic behind their reasoning in these scriptures in view of the fact that the other three time Priscilla and Aquila are mentioned in scripture, Aquila is named first (CP Ac 18:2, 26; 1 Cor 16:19).

Rather than interpret scriptures to suit our agendas, we should accept what they teach at face value: Priscilla simply assisted both Aquila and Paul in their ministries. This accords with what other scriptures teach against women being leaders over men in the church (CP 1 Ti 2:8-15; 3:1-7; 3:8-13).

See also comments on Ro 16:1-2, 1 Cor 14:34-35, 1 Ti 2:8-15, 3:1-7 and 3:8-13. and author's studies The Church and Women and God's Order for the New Testament Church in his book Advanced Studies in the Christian Faith (Volume 1).

Acts: