How at least one or more contemporary theological development (Liberation, Feminist, African/ black theology, or Historical Jesus’ Scholarship etc.) can be related to Pentecostal Patristic doctrinal development - whether by criticising it, confirming it or elaborating it.
Black theology as expressed in Black Pentecostal churches that were organized in the early twentieth century following the Azusa Street Revival confirms Patristic doctrine of missiology (pertaining to the mandate for missionary work). Black theology Pentecostal churches developed and continue to support a strong missionary presence in places like Africa, the Caribbean, Asia, and South America.
The doctrine of the Holy Spirit in the early church has a interesting development. In many ways, the Holy Spirit was the silent member of the Trinity. The Father and son took most of the focus of the church. However, as time progressed, the doctrine of the Holy Spirit came to focus. In short, the Holy Spirit is of the same substance of the Father and Son, that is, co-equal in every sense. In light of this, within the unity of God, there was also diversity. This idea of diversity has been emphasized by nearly all modern theologians.
I have to agree with accessteacher. Outside of how some feminists have identified the Holy Spirit as feminine, and some even as God as feminine, I cannot think of how the Patristic period has impacted modern religion.
In response to #2, interestingly some feminist theologians have seized upon the idea of the Holy Spirit and used it to promote gender equality, especially as regards the idea of the Holy Spirit acting as a more feminine and empathetic force than the male Jesus and the patriarchal image of God.
It would be helpful if you could give us a more precise idea of what you mean by pentecostal patristic doctrine. Are you thinking of the doctrines of so-called "Pentecostal churches"? Are you thinking of medieval patristic doctrines concerning pentecost? Knowing more clearly how you are defining this term would help us give you useful responses.
Feminist theorists would perhaps argue that the very idea of a patristic church founding (the "church fathers" influential in the articulation of early Christian thought) were an example of the ways individuals attempted to use religion to establish gender norms. There were women who wrote on Christianity in the three centuries or so after Jesus's death, and their writings are not part of the patristic canon.