Sorry for monster post.
The "fully involved women" are the women disciples and followers of Jesus who are named throughout the Gospels and Acts.
Mary, Jesus' mother. Probably present through most of his ministry - certainly there at the first miracle at Cana (and prompted it), and at the crucifixion. Named as present at Pentecost too.
Mary Magdalene. Described in Luke (8:1-3) as having being cured of seven demons and as one of the group of women who were travelling from town to town with Jesus and the Twelve through his ministry and supporting his ministry. Present at the crucifixion, burial and the first witness to the resurrection. Told the Twelve that Jesus had risen.
Salome. Present at crucifixion. Probably also the woman who petitions Jesus for her sons to sit with him in Paradise. Mentioned as at the tomb in Mark.
Mary the mother of James. Named as one of the women who saw the crucifixion and who went to the tomb afterwards.
Joanna - another women who had been healed and travelled with Jesus, financially supporting the group. Wife of Herod's steward and so possibly an eyewitness for Luke the Evangelist on the court of Herod. Another of the first group of witnesses to the resurrection.
Mary of Bethany. Sister of Lazarus. Treated as a disciple as discussed above.
Martha of Bethany. Sister of Lazarus. Jesus also treats her as a disciple after the death of Lazarus, teaching her that he is the resurrection and the life.
Susanna - also healed and travelling with the group and supporting them.
Mary of Clopas - present at the execution. Possibly she is also the unnamed disciple the risen Christ appeared to on the road to Emmaus.
Tabitha/Dorcas - named in Acts. Peter raised her from the dead.
Lydia - named in Acts. A merchant converted by Paul who had the apostles stay at her home.
Mary the mother of Mark John - named as hosting as housechurch in Jerusalem.
Damaris - named in Acts. A Greek woman converted by Paul in Athens.
There were probably more unnamed female followers and disciples, and many of the women who were present at the crucifixion (described as the group who had followed from Galilee) were probably also present at the entry to Jerusalem and the Last Supper.
Plus there is Peter's MIL mentioned in Mark as having been healed by Jesus.
Women named by Paul:
Priscilla. Named in Acts, Romans, 1 Corinthians and 2 Timothy. Priscilla is always named with her husband Aquila - they seem to have been a co-missionary couple who taught theology together. Paul lived, travelled and worked with them and it is clear that he regards them both as co-workers - half the time he names her first and specifically calls her a co-worker. They were Jewish tentmakers who met Paul in Corinth, and were probably involved in the founding of the church in Corinth along with Paul. In Romans he says that they both risked their lives to save his. In Acts they are described as having taken a preacher to one side and having taught him better theology. Possibly the author of Hebrews.
Junia. Called an apostle (in fact 'outstanding among the apostles') by Paul in Romans.
Euodia. Called a co-worker by Paul in Philippians.
Syntyche. Called a co-worker by Paul in Philippians.
Phoebe. Mentioned in Romans as a deacon in the church in Cenchreae, and as a benefactor to Paul.
Julia. Named in Romans.
Nympha. Named in Colossians as having a church in her house in Laodicea.
Apphia. Philemon is addressed to her.
Tryphena, Tryphosa and Persis - mentioned as workers in Romans.
Chloe - mentioned in 1 Corinthians. 'Her people' tell Paul that there is dissension in the church in Corinth, which probably means she leads a church there.
Then there's the unnamed women present at Pentecost, who probably included the group who had seen the crucifixion and resurrection, and the unnamed women who are eating with the men after Pentecost.
Also the wives of Peter and the apostles mentioned in 1 Corinthians.
The passage about women teaching in church is from 1 Timothy which is one of the deutero-Pauline letters. It's certainly not consistent with Paul's attitude to and relationship with Priscilla, and there are other reasons for not considering it to have been genuinely authored by Paul.
And going back to my original point, this all adds up to women being involved in the life of the early church in a way they wouldn't ever have been in the contemporary Jewish or Helleno-Roman world.