Answer: The most fundamental, root belief of Islam is that 'there is no god but God'. We believe that there is only one God, the Creator and Sustainer of the heavens and earth and everything in between. He governs and directs all affairs, and all matters are referred to Him for decision. God is independent and self-sufficient and the creation is dependent on God for everything.
The complete first statement of faith that you referred to is: 'there is no god but God, and Muhammad is the messenger of God'.
A detailed statement of faith, known as Iman-e-Muffassal is: 'I believe in Allah, His angels, His books, His messengers, in the last day, in predestination: that all good and bad is from Allah, the Most High, and I believe in resurrection after death.'
We humans have been sent to Earth for a limited time and for a purpose. We humans are repeatedly tested in this lifetime, and every fine detail of our lives is recorded.
We have been endowed with free will so that we can choose our thoughts and actions. However, this does not mean that God is unaware of what we will choose to think or do. Human free-will and God's complete knowledge are not mutually exclusive. It is God who decides who to bless with faith, because only God knows our hearts.
Regarding prayers, we need them. If we pray, it is to our own benefit. If not, it is our own loss.
Similarly, when we choose to sin, whether it be against God or against humanity or against creation in general, we are actually being unjust to ourselves.
God is not unjust towards creation, rather God is loving, merciful and kind.
It is the duty of messengers and all those who learn from them to convey the message of God to others. Whether others choose to believe or not is between God and each individual soul. God is oft-forgiving, and loves to forgive. To quote a verse: Quran [2:186] 'And when My servants question thee concerning Me, then surely I am nigh. I answer the prayer of the suppliant when he crieth unto Me. So let them hear My call and let them trust in Me, in order that they may be led aright.' [translator: Pickthal]
There is no compulsion in religion, God does not impose faith on anyone. God knows best our hearts and our souls.
We believe we have a common history with Adam, Noah, Abraham, Moses, Jesus and all other messengers of God. We believe they all preached the same faith in one God, resurrection and the last day. Muhammad was the seal, the last of the series of prophets.
As opposed to the Christian concept of Jesus atoning for humanity's sin through his death, we Muslims believe that each human is responsible for his / her own self. Nobody can bear the burden of another. Each one of us will be judged and recompensed according to our own faith, beliefs and deeds. We do not believe that we are here on Earth because of Adam and Eve's sin. As I mentioned a few posts ago, the purpose of Adam's creation was a 'khalifa' of the Earth, i.e. a vice-regent or someone who is responsible for the Earth. We were predestined to come to Earth for a temporary abode. The resurrection is also predestined, and so is a permanent life thereafter.