The name “Presbyterian” applies to a diverse group of churches that adhere in some degree to the teachings of John Calvin and John Knox and are led by representative elders (presbyters) of their congregations. Here is a brief overview of the history of the Scottish Presbyterian Church:
The Presbyterian Church was first organized in Scotland under the leadership of the Reformer John Knox. The Church of Scotland was affiliated with the Roman Catholic Church, even though it maintained an attitude of independence. John Knox was a priest in the Church of Scotland and was fed up with the abuses he saw in the Catholic leadership. Knox was exiled to England after his involvement in the murder of Cardinal Beaton in 1546. While in England, he was licensed to preach in the Church of England and was instrumental in reforming the Book of Common Prayer.
When Mary Tudor ascended the English throne and started her bloody persecutions of Protestants, Knox fled to the Continent, where he met John Calvin and began to study Reformed theology. In 1559, Knox returned to Scotland and became a vocal proponent of Reformed theology and the concept of Presbyterian leadership in the church. A number of Scottish lords had already been promoting religious reform, and they gladly supported John Knox’s teaching. Under Knox’s leadership, these “Lords of the Congregation” wrote the Scottish Confession of Faith in 1560. This confession ended papal rule in Scotland and outlawed the Mass. The Scottish Confession remained the primary doctrinal guide for the Church of Scotland until the Westminster Confession of Faith in 1647.
The article goes on to explore the development of the Presbyterian Church and contains links to other useful, related articles. The article concludes:
Within the broad category, there are some which can be considered conservative or fundamental, and some which would be called liberal or progressive... While most Presbyterian churches will agree on general themes such as the depravity of man, the holiness of God, and salvation by faith, there is wide divergence in how they define and apply those themes. Source: https://www.gotquestions.org/Presbyterians.html
With regard to the three specific questions you have asked, I can only help on the first question: What are the main beliefs of Presbyterianism? Below is a brief summary of what the Free Presbyterian Church of Scotland believes, “expressing our adherence to all the fundamental doctrines of Reformed, Calvinistic (not merely “Evangelical”) Christianity.”
Scripture. The Bible is the fully inspired, infallible and inerrant Word of God, and therefore the only supreme rule for faith and life.
God. There is one God, who decreed all things from eternity, created the entire universe in six days, and governs the whole creation continually.
Trinity. This one God exists eternally in three Persons: Father, Son, and Holy Spirit.
Man. Man was originally created in the image of God, and his main duty in life is to know, love, glorify and enjoy God.
Sin. All fell with Adam in his first transgression. Everyone is guilty of sin (which is disobeying God’s law), and deserves to be punished in hell for ever.
Jesus Christ. God in His mercy sent His only begotten Son to be the Saviour of sinners. By taking human nature into union with His divine Person, Jesus Christ is both God and man. He was born of the virgin Mary, lived a perfectly sinless life, died on the cross, rose bodily from the dead, and ascended into heaven.
Atonement. When Jesus Christ died on the cross, He was punished in the place of His people for their sins, as the only way to satisfy divine justice.
Faith. Those who trust in Christ alone for the forgiveness of their sins, and repent, are accepted by God and will go to heaven. No one can be saved in any other way.
The Holy Spirit. Only the Holy Spirit can bring people to faith and repentance, which He does when He regenerates them into spiritual life.
The Second Coming. Jesus will return bodily at the Day of Judgment, when everyone who died will rise again. Those who trusted Him for salvation will go to heaven with Him for ever, but the rest will be cast into hell to endure conscious torment for ever. Source: https://www.fpchurch.org.uk/about-us/what-we-believe/
That article also provides a link to a comprehensive statement of their beliefs based on the Westminster Confession of Faith: https://www.fpchurch.org.uk/about-us/important-documents/the-westminster-confession-of-faith/
To know what makes them different from other churches , see What We Contend For: https://www.fpchurch.org.uk/about-us/what-we-contend-for/
Finally, my only other comment with regard to your first question is that there should be no need for any person to pass themselves off as a member of a particular denomination. Personally, I self-identify as a Christian, first, foremost and last. I don’t belong to any denomination. I belong to Christ Jesus and am a member of His Church. I sincerely hope this helps as you set out on this interesting and important voyage of discovery.