Philosophy and Proofs
Atheism includes any philosophy which claims that no God or gods exist, including any rational or reasoning creator of the universe.
Christianity is the philosophy that claims that Jesus of Nazareth is the Son of God and Jewish Messiah (or Christ); that he was crucified, died and was buried, and rose again three days later; that his death was an atoning sacrifice for the sins of his followers; and that placing one's trust in Jesus is how any sinful person can be forgiven and made righteous before their creator.
Deism refers to any philosophy which claims that there is a supreme creator of the universe who has no ongoing involvement with what is created apart from the initial act of creation itself.
Empiricism describes any philosophy which claims that all knowledge originates in experience, denying the validity of both deductive reasoning and divine revelation.
Islam refers to the specific philosophy of the Muslims, a monotheistic and unitarian belief system declared by Muhammad in 610 AD and described in the book known as the Qur'an.
Monotheism refers to any philosophy which claims that there is only one supreme creator of the universe.
Open theism refers to any philosophy which claims that there is a supreme creator of the universe who does not foreknow the outcome of human choices.
Polytheism refers to any philosophy which claims that there are multiple supreme creators of the universe.
Rationalism describes any philosophy that claims that beliefs and opinions should always be logical, deductive conclusions rather than being based on experience, observations, religious teachings, or divine revelation.
Theism includes any philosophy which claims that some kind of God or gods exist.
Trinitarianism is the philosophy that there is only one God in terms of essence or being; and that the one God is revealed through three eternally distinct persons, specifically, the Father, the Son (Jesus of Nazareth), and the Holy Spirit. Trinitarianism is unique to Christianity.
Unitarianism refers to any philosophy which claims that divine sovereignty is not shared in any way.
In Printed Form
Along with numerous other authors including Don Landis, Bodie Hodge and Roger Patterson, Timothy McCabe contributes analyses of various world religions and cults in this volume from Master Books.
"What is an easy way to demonstrate that open theism is false?"
Open theism is the belief that god exists, but does not know the future outcomes of human choices. Instead, he finds out about them when they happen. Open theism renders all human conclusions completely irrational. A god who does not know what the future holds clearly did not design the future. And if he did not design the future, then there is no good reason to believe that it must behave in the way he wants it to behave.
"Why does God change His mind so much? (Gen 6:7; Exd 32:14; Jdg 2:18; 1 Sam 15:35; Jhn 3:10 etc)"
Below are the verses cited in the question. John 3:10 doesn't seem to fit the question, so I assume it was a typo. Nonetheless, the other verses should more than suffice to make the point the questioner intended. Genesis 6:7 The LORD said, "I will blot out man whom I have created from the face of the land, from man to animals to creeping things and to birds of the sky; for I am sorry that I have made them.
"Why am I here? Why do I exist? What is my purpose?"
According to Christ, the greatest commandment is to "love the Lord your God with all your heart, with all your soul, with all your mind, and with all your strength" (Mark 12:30). As our greatest command, it is the thing we most ought to do. We exist to love God. How are we to love God? What exactly does that mean? The apostle John tells us that love for God is to obey Him (1 John 5:3).
"Should modern mainstream religions be considered cults?"
Modern mainstream religions could be considered "cults", and every form of Atheism could as well. But why should it matter if that particular term can be applied? Why would we want to call every modern mainstream religion a "cult" apart from a desire to provoke some type of defensive emotional reaction from religious persons? And why would anyone be interested in manipulating such a reaction?