Peace CyberAnthro!
You raise a good topic.
I will address the topic in three points.
1. All languages mutate over time, including Arabic. This is simply a semantic given. However, the more important question is "Can we trace the original meaning of a statement in a language which is no longer used?" For some dead languages, the answer is "No," because there are no extant texts authored in that language from which we could derive an understanding of the manner in which words were used. In other dead languages, the answer is "Yes," because there are sufficiently enough texts in existence to allow us to garner an understanding of how its speakers used their words. Fortunately, Arabic was preserved in writing from the 9th century onwards. And it wasn't simply preserved by several books. It was preserved by a massive body of texts dating back to that time. The only concerning factor is that there are very few preserved texts (not including the Quran) that precede the 8th century. So we do have a gap in our evidence lasting for a couple of centuries. However, during this time, the Quran would have served as a common denominator amongst all the Arabian tribes after having accepted Islam. This, in my mind, would have certainly contributed to the preservation of spoken Arabic from the 6th century to the 9th century. This is especially true amongst the bedouin Arabs because their preservation of history and lineage was almost entirely oral, and they placed a great emphasis on oratory. These two factors do not negate the liability of mutation of Arabic throughout the 6th and 7th centuries, but they do go some way in abating it.
2. I like to distinguish between the Arabic used in the Quran, and the Arabic used by people after the Islamic expansion in the 8th and 9th centuries. I prefer to call the Arabic used in the Quran "Peninsular Arabic." Whereas, I prefer to call the Arabic used by people post-750 AD "Classical Arabic." I use the term "Peninsular" because Arabic was confined to the Arabian peninsula at the time of the Quran's revelation, whereas "Classical Arabic" was used throughout the Middle East, North Africa, and even parts of South Asia. I make the distinction between "Peninsular" and "Classical" because there were many new terms introduced to the language after circa 750 AD. This was a response to encountering foreign cultures and it was an attempt at Arabising foreign concepts; with particular regard to non-Arab Christian works and the works of Classical Greek philosophers. I have little doubt that new words were added to the language, but this in no way implies that the meanings of original words were forgotten, if anything, it strengthened them. There are many examples of morphology I could give where this is the case, but I feel that my post will be too long for the casual reader.
3. The available evidence relating to the use of Peninsular Arabic is sufficient to the extent that we may accurately understand the intended meaning of the One who authored the Quran. I do not encourage people with an elementary understanding of Arabic to read the Quran in its original text. Otherwise, in my view, an elementary student will do more harm to their understanding than they will aid it. You need to know your stuff if you want to read the Quran in its original form. I always encourage people to read several different translations of the Quran in their mother tongue, whether it be Urdu, Punjabi, Mandarin, French, Dutch etc. Using several different translations will allow the reader to even out any biases that the translator consciously or unconsciously imprints in the text.