With regard to proof of the existence of Allah, it is obvious to anyone who ponders the matter, and there is no need for a lengthy discussion. When we ponder the matter, we find out that it is divided into three categories: instinctive evidence, tangible evidence and from my religion which you wouldn't care.
1 – Instinctive evidence: The instinctive evidence that God exists is the strongest of all evidence for those who are not led astray by the devils. Hence Allah says (interpretation of the meaning):
“So set you (O Muhammad) your face towards the religion (of pure Islamic Monotheism) Haneef (worship none but Allah Alone). Allah’s Fitrah (i.e. Allah’s Islamic Monotheism) with which He has created mankind” [al-Room 30:30]
Man’s sound nature (fitrah) testifies to the existence of God and man cannot turn away from that unless the devils mislead him; whoever is misled by the devils may not recognize this evidence.”
Every person feels inside himself that he has a Lord and Creator, and he feels that he is in need of Him; if some major calamity befalls him he turns his hands, eyes and heart towards the heavens, seeking help from his Lord.
2 – Tangible evidence: This refers to the things that exist in this universe; we see around us things that exist, such as trees, rocks, mankind, the earth, the heavens, seas, rivers…
If it is asked: these things are so many – who created them and is taking care of them?
The answer is that if these things came into being by accident, spontaneously and with no cause, then there is no one who knows how they were created, and that is one possibility. But there is another possibility, which is that these things created themselves and are taking care of themselves. And there is a third possibility, which is that there is Someone Who created them. When we look at these three possibilities, we find that the first and the second are impossible. If we reject the first and the second, then the third must be the one which is correct, which is that these things have a Creator who created them, and that Creator is Allah. This is what is stated in the Quran, where Allah says:
“Were they created by nothing? Or were they themselves the creators?
Or did they create the heavens and the earth? Nay, but they have no firm Belief” [al-Toor 52:35]
Moreover, when were these mighty things created? For all these years, who is it that has decreed that they should remain in this world and has granted them the means of abiding?
The answer is, it is Allah who has given to each thing that which is suited to it and will guarantee its survival. Do you not see the beautiful green plants; when Allah cuts off their water supply, can they live? No, rather they become dry stalks. If you ponder all things you will find that they are dependent upon Allah. Were it not for Allah, nothing would remain.
And Allah has created everything to do that for which it is suited. So camels, for example, are for riding. Allah says (interpretation of the meaning):
“Do they not see that We have created for them of what Our Hands have created, the cattle, so that they are their owners.
And We have subdued them unto them so that some of them they have for riding and some they eat” [36:71-72]
Look at the camel and how Allah has created it strong, with a strong back, so that it can be used for riding and it is able to endure harsh conditions which other animals cannot bear.
If you look at other creatures you will find that they are suited to the purposes for which they were created. Glory be to Allah.
“All the divinely-revealed laws point to the existence of the Creator and to the perfect nature of His knowledge, wisdom and mercy, because these laws must have been prescribed by someone, and that Lawgiver is Allah.”
With regard to your question: why did Allah create us?
The answer is: so that we would worship Him, thank Him and remember Him, and do that which He has commanded us. You know that among mankind there are unbelievers and there are Muslims. This is because Allah wants to test His slaves as to whether they will worship Him or worship others. That is after Allah has showed the way to everyone. Allah says (interpretation of the meaning):
“Who has created death and life that He may test you which of you is best in deed” [67:2]
“And I (Allah) created not the jinn and mankind except that they should worship Me (Alone)” [51:56]