Prophet Mohammed could read and write; however the level of his education was not enough as to be considered educated or literate. Civilization began in Egypt being the first world power to rule the earth; followed by Assyrian, Babylon and Greece. The oldest university on earth was founded in Egypt by Alexendar the Great of Greece as the center of leaning. These ancient world powers influenced the Middle East region a great deal in the act of reading and writing being the centre of the accient world powers. Therefore during those accident period the Arabic reading and writing was introduced as part of their cultural heritage which is used as a means of communication among Arab nations.
Therefore Prophet Muhammad was born to this highly educated Arab community, was fory years of age before he wrote the Koran and was involve in trading activities as part of his education and as well as involving in reading and writing because of his environment although not highly educated like others due to the fact that he had no body to bring him up to the lime light in terms of Arabic education but God used his little idea he had attained in reading and writing and brought him up to the lime light scriptural like many other preachers and founders of religious organization we know today. By his earlier associating with educated Arabs both at home or in trading activities, he also got the idea of reading and writing on a very slow space but not too sound because there was none to train him to a considerable level since when he was young. In other words, he did not attend an Arabic School to considerable level as he lived with other people who were not his real parent and who may occupy him more of the domestic works just as it is common even in this modern day.
Through other people he associated with, God gave him the grace to read and write although with errors in the Arabic presentations and as well as reading on a slow space although he could speak Arabic very fluently. God performs miracle with what you already have at hand and then increase it to a higher level. For example God used the staff of Moses in his hand as a shepherd. Little David killed the Goliath with a sling and a stone because he was use to it as a boy and not even the armour, sword and the javelin of King Saul. The widow little flour and oil at hand was increased for there and half years when she first of all fed Prophet Elijah. Elisha his sevant increased the little oil of the widow and filled all the empty pots which she sold to pay off her debts. Jesus fed Five thousand people with the five loaves of bread and two fishes that was available with a sales boy in the field etc. Similarly God only use you to preach with the language you can freely speak except you lean any other; and will use you in your own hand writing, spelling etc depending on the level of your education. On the day of Pentecost the Apostles spoke in diffidence languages even spoke in Greek to the hearing of others miraculously; amd it stopped that day while they used their original language to preach about the Good news because that speaking in Greeek did not flow again was temporally. If they want to continue speaking in Greek then they have go to Greek school and lean it. For the fact that anything supper natural does not last too long; it comes occasionally. Even Jesus Christ perform his healing whenever the power to heal was with him so the bible said.
So you cannot misuse the power of God anyhow. Although he gives it to you yet he controls it from above. Samson fought his enemies whenever the spirit of God descended on him because he could not perform on his own even though he had the seven logs on his head. Thus it is written" Is not by might nor by power but by my spirit says says the Lord" So the God of Abraham works in the mist of his prophets both of Ishmael and Isaac descendants and those that share the Abraham faith such as those in Judaism, Christainity and Islam.
For example, Prophet Jeremiah was one of the greatest Hebrew prophet whose prophetic power took Israel into Captivity in Babylon. Although could read and write in Hebrew or Aramaic but the level of his education was not too sound as to present a letter to King Joachim of Judah so he employed Baruch who wrote for him at his dictation. God did not hold prophet Jeremiah's hand to write correctly as a prophet; if Jeremiah was to write, he could only write based on his limited knowledge and nothing more nothing less because that is what he could afford to write (Jeremiah chapter 36).
The same thing happed to Prophet Mohammed who could write down those visions and Messages from Angel Gabriel based on his limited Arabic education which will be subject to errors and presentations in the Arabic format. But as God blessed his ministry with some educated Arabs, they now carried out the editing work of his writings so as to be presentable to the educated Arab community. But if you try to defend that he could not even read or write, then it means other educated Arabs wrote for him who may even change the words to suit themselves in order to counter Judaism and Christianity to an extend; above all, Prophet Mohammed could not even detect such errors or changes because he could not even read what others wrote in his name therefore loose his authorship. That is the overall implication. But the truth of the matter is that he could read and write at his level when he wrote the Koran as the sole Author.
Some people are good in speaking and reading but they could not write very well even no matter the high level of their education but when you put them to test you will be surprised! For example Apostle peter was a fisherman who had little knowledge in the Hebrew education for the fact that the Jews encourages their kids in early education within their religious set-up so that they too can be able to read the Torah known as the Old Testament scriptures.
On the days of Pentecost, Apostle Peter spoke like a university graduate as he was quoting from the old testament scriptures perfectly; so the Jews were surprised who asked "who taught this people when they have not leaned" Despite his popularity he could write very few books and were edited by other Apostles who were more leaned than himself; while the educated ones like Apostle Matthew, John and Apostle Paul wrote more books due to the level of their education.
Writing the Holy Koran was not a day's job; the first is the editing stage which will required educated Islamic scholars who were direct assistance to the Prophet and his secretary. After the editing, the Prophet has to proof read them to ensure that there was no errors or changes since he could read in Arabic as the Chief Editor page-by-page and then finally approved for publication. So that the content in the Koran remain intact.