For the most part, yes. Latin is one of the most studied languages in history, and we have a surprisingly large number of written and translated works from the Romans that we can base our understanding of he language off of.

One thing that would sound very strange to the average Roman person would be the tendency to use the phoneme "ch" in place of hard c's. This came into practice becaise of modern Italian, which, while being a highly related language, does have significant differences. For most academics, this practice has fallen into disuse in recent years.

Unless the modern speaker was extremely proficient, sentence structure would also be very strange compared to the average historical Latin speaker. Remember that Latin has very little need for firm sentence structure, as the endings of Latin words relate them to each other more than their position relative to the other words. And while this practice may be more prominent in "high-brow" Latin, subordibate clauses were far more prominent in the language than they are in modern vernacular.

I have a side question: did Latin ever incorporate loan words from other languages spoken throughout Europe at the time, as modern English does (i.e "tornado", "mosquito", etc)?

I should think much would depend upon the class and time period of the ancient native speaker!

But, having studied both classical and ecclesiastical Latin, I suspect it'd be more difficult for academics to understand clergy and vice versa than for certain ancients to understand modern Latinists!

The pronunciation is so different (as articulated in other comments here)! Now that I've been used to Church Latin for just over a decade, it's actually quite a struggle for me to comprehend listening to Classical Latin, despite having started out there.