Freja, you said Islam is rooted in Arab culture, which I'm afraid is a wrong understanding. The last Prophet of Islam, Muhammad (may the peace and blessings of Allah be upon him), was an Arab, but not all were. Islam is not linked to a specific culture. I'll approach the prayers in Arabic issue afterwards. Islam is not a new innovation, but quite the contrary - it was what has always been, the original and true and natural religion and way of life, ordained for man by God.
Allah has sent a succession of prophets to people throughout the ages, to convey His message to them, and with guidance to show people how He wants them to live and worship Him. Muslims believe in all the prophets Allah sent, and do not reject or blaspheme any of them, from Adam, to the last and final prophet, Muhammad, peace be on them all. They were the purest and noblest of humanity and were not divine in any way. Allah sent all the prophets with the same message and not different messages. All the prophets and messengers sent by God to humans to convey His message, conveyed the same message since the beginnig of time, regardless of when and which people they were sent to. The message was: to submit wholeheartedly to Allah and worship Him and Him alone, without any associates in, or parts to, His Exclusive Divinity, and to obey the prophet. They taught that people should be under no misperception that they can commit themselves to Allah as their Lord, and then combine this with accepting others as their Lord, or associating others in His Divinity, in whatever way. They taught that we should strive hard to translate our belief in the One True God into practice, by obeying Allah and the messengers He sent, who were also role models and examples for us, showing us practically how to put the guidance they were sent with into practice in our daily lives, explaining the scriptures, warning against wrong-doing, giving good tidings, and giving additional legislation from Allah.
So Islam is not a new faith, but is the same ultimate universal truth that God revealed to all the prophets, including Noah, Abraham, Moses and Jesus (peace be on them), and the same thing they all taught. Thus Islam is not named after a specific person (like Christianity, Buddhism), nor after a certain race or place (like Judaism, Hinduism), but is named by God Himself, the meaning loosely translating as 'submission to God', which is what every Prophet and their righteous followers did, from amongst all times, places and peoples. That in itself is one fraction of the evidence that it was the way of all the Prophets from the beginning.
With time, the message got forgotten or corrupted. So people started worshipping other gods along with Him, made idols, said that God begot a son, said that certain people were incarnations of God, some rejected belief in God altogether, while others elevated the status of some prophets to divine, or at the other end, rejected or blasphemed some of the prophets. Whenever God's message got distorted by people, or forgotten, a new messenger was sent, not with any new or changed message, but reinforcing the actual message that God sent all the messengers with, the actual core beliefs that people were taught from the beginning of humanity, confirming the true parts of previous teachings and scriptures, and correcting wrong beliefs and misconceptions that had crept in. God required that whenever He sent a new messenger, that messenger should be followed along with any new scripture given to him. This chain of messengers culminates in prophet Muhammad (peace on him), who is the last and final prophet and messenger. Since his prophethood, God's message is available unchanged and unadulterated, for the entire world, until the end of time. He wasn't sent as prophet and messenger for a specific group of people and specific time (e.g. as Moses and Jesus were to the Children of Israel), but he was sent for all the world, for all time, until the Day of Judgement. Thus he is the last, not first, prophet of Islam; a messenger to all mankind, for now, and for all time to come. He is the messenger who must now be followed.
When Jesus (peace be upon him) was sent to the Children of Israel, the language of the scripture that God gave to him, and worship, was most likey in Aramaic. Prophet Muhammad (peace be upon him) was the last and final messenger sent for all mankind. He was born in Arabia, at a time when the Arabs were submerged in polytheistic culture, and badly needed reforming, and that place was therefore the hub from which Islam spread. Allah laid out the rules of worship, which include certain worship which must be in Arabic, and certain that can be in your own language. This is not because the culture is important, but because that was the language that Prophet Muhammads first followers spoke. If they didn't understand what was being sent to them, what is the point? Imagine sending them a book or instructions in Chinese! Having one language for the Qur'an and for certain acts of worship means when we read the Quran we all, whatever country we're from, recite the same thing, word for word. Any mistakes if someone makes them, can easily be spotted. And it's such a great feeling for universal brotherhood when you know you all pray the same way and read the same words, which, incidentally, are Gods words, not the words of any human.