Most Arabs don't speak Modern Standard Arabic either; they speak dialects. Educated Arabs learn MSA, but it is more of a literary language. For more information on that:
Diglossia - Wikipedia, the free encyclopedia
Moving onto the difference between MSA and Classical Arabic: there isn't much difference other than in vocabulary. MSA is a direct descendant of the latter. Therefore, people who know MSA should be able to read through the Qur'an with little difficulty. For more information on Classical Arabic:
Classical Arabic - Wikipedia, the free encyclopedia
Classical Arabic and MSA are written in the same alphabet, which yes, is a consonantal alphabet. I don't disagree with anything else you have said.
I think you're confusing word roots and the use of vowels.
Let's take 'writing'. I'll use Wikipedia's provided transliterations and type some of the words in Arabic so you can see it for yourself.
Source:
Classical Arabic - Wikipedia, the free encyclopedia
He wrote (kataba):
كتب
He writes (yaktubu):
يكتب
Writing (kitaabah):
كتابة
Writer (kaatib):
كاتب
The problem with vowels is that the short vowels were not written down (a, u, and e to use rough approximations), and usually aren't (exceptions nowadays are in Arabic books for children, situations where words might be confused). Long vowels (aa, oo, ee) were always written down, as far as I know, otherwise we would have a much harder time of figuring out things.
For a specific problem related to the use of vowels using the verb 'to write', let us take the word 'kataba'. The a's are all short vowels, so what is written is three letters: k-t-b. Most of the time, the odds are the word is referring to kataba. However, how do you say 'someone wrote'? Kuteba. (If I remember correctly.) Since those are all short vowels, all you see is k-t-b. So how can you tell the intended meaning? The use of diactrical markings (to show short vowels) or by context.
Most of the time, it's not an issue.
I hope this post was of some use.