What are the essential beliefs of orthodox Puritanism?

Answer 1

First of all, "orthodox Puritanism" is an oxymoron.
But puritanism sought to rid the church of what it perceived to be the "Catholic additions" to Christianity, to "purify" it back to what it originally was in the early centuries. Of course, their ideas of what constituted early Christianity weren't based on anything historical, but on a Calvinist doctrine of sola scriptura (only the Bible).
So the essential belief was a strict adherence to (calvin's interpretation of) the Bible. Outwardly, this entailed austere forms of dress and liturgy. But it also meant an emphasis on personal piety, prayer and devotion, and Bible study.

Answer 2

Puritan Orthodoxy

Answer 3

A Puritan of 16th and 17th century England was any person seeking "purity" of worship and doctrine, especially the parties that rejected the Laudian reform of the Church of England. Those who sought further reform of liturgy and theology away from that of the Roman Catholic Church and those who justified separation from the Church of England following the Elizabethan Religious Settlement are commonly called "Puritans" by historians and critics. Later groups are called "puritan", not necessarily favorably, by comparison to these low church Anglicans and Calvinistic Non-conformists.