Finally something I am qualified to answer! Please excuse my English, as I am not a native speaker. So the idea of Sanskrit words having power has two sides, the historical and the (probably more fun) mythological.

Historically, the most revered Sanskrit texts are the four Vedic Samhitas. While the later parts of them are not so hard to understand as they are close to classical Sanskrit, the oldest parts are incomprehensible. During my studies we had to translate parts, and while out of some you could cobble (with some effort) a coherent sentence, some parts were so archaic that it was pretty much impossible to translate, one could only guess. Now the thing about the Sanhitas is that they are treated with such reverence that is it forbidden to change even a syllable, and there are techniques where the words and syllables of the verses are spoken is various sequences (the eleven Patha and Swara method) to rule out deletions, interpolations ormodifications. As you can guess this is strongly connected to the belief that not only is the content of these texts of the outmost importance, but also the sounds.

These texts are dated by conservative scholars to ca 2000 BC, and even older by others. Now, already in the past these texts could not be easily understood, and many efforts had been made to provide glossaries and commentaries to them. Not many have survived, the ones that did were written in times linguistically far removed from the language of the Sanhitas. The most famous one of them was the commentary written by Sayana, it set the tone for the next hundreds of years until the present time. Now Sayana was a ritualist, and tended to explain verses in a way that they fit their ritualistic use. (During these time the Sanhitas were used spells or invocations or blessings in rituals of all kinds). Where a verse was too difficult to translate, he would find a word he understood, like "home", and state this verse is meant to bless the home.

Now, this is tied up to the class struggle between the priest and the warrior class. In the Vedic Sanhitas the castes seem to not yet be hereditary, and householders performed their own rituals (offerings to the home fire). In the later texts the rituals grew more and more complex (the Brahmana texts describe some that take days, where hundreds of animals were sacrificed, involving extremely pedantic rites); and have to be performed by specialists as even a small mistake could not only render the entire ceremony useless, it could bring negative consequences. These mistakes include mispronouncing a sacred verse. Thus the priests took over the role of the go-between between men and the gods, and even the ruling class was dependent on them in this matter. The priests underwent rigorous training which included memorising entire Vedic Sanhitas.

With this background it is pretty understandable why Sayana holds that the Vedic texts, even if they cannot be understood, and even if they don't have a rational meaning still have (supernatural) effects. The consequences of this belief, as well of the divide between the castes were rather dire. The lawbook Manu Smriti states that if someone of a lower castes eavesdrops on the Vedic texts, molten metal should be poured into the ears, if he repeats them them molten metal should be poured into his mouth. This points that during Manu's time the Vedic Samhitas were seen as something meant only for the upper castes, and had a power or holiness which made them unsuitable for those of lower birth. This draconian point of view fortunately did not survive till today. I am not going to go into detail about the Hindu Reform Movement, I will just mention that it has brought the ancient texts back to the "common man". The belief that prevails is that the Vedic texts as well as later Sanskrit prayers, stories and other verses are holy or positive by themselves, and they have a positive effect when repeated, even if not understood (possible parallels with Christians praying or hearing mass in Latin).

Now, back to the power of words and the class struggle between the priests and the warriors, lets take a short look at the Indian mythology and epics. One of my favourite ones, the Mahabharata, has several interesting bits that show the belief of the power of words. The Rishis (high ascetics) were able to curse or bless with just a few words, and these curses always came true. Even a misspoken word, or something said by accident would come true, if said by a person of authority (a priest, a parent or even older sibling) in the form of a blessing or curse.

In the battles of the Mahabharata "astras" or very powerful weapons played a big role. These weapons were described as being extremely devastating, hence the comparision to thermonuclear weapons. Now, astras were not weapons per se, rather they were a power that could be invoked into a weapon. So a warrior would loosen a regular arrow from his bow, but before that he would invoke the astra. Each astra had a specific effect, most were extremely powerful, were practically unstoppable, and many had very specific rules about how it could be used. The knowledge of these astras would be passed on from teacher to student, or could be bestowed by a diety. The astras were invoked with specific words. However from the context it is pretty clear that it was not the mere words alone that invoked the astra, the warrior had to have the "power" to use it bestowed upon him.

Another interesting topic in sounds as symbols. Om, the sound out of which the entire universe is supposed to be created. Or the sounds that corresponds to the different chakras (energy centres) of the body. Now this is unfortunately not my field.

To add a bit to your questions -- Unfortunately many Sanskrit students (and even teachers) in India are pretty much cut off from a lot of information about the world. Statements like Sanskrit is the oldest language in the world are unfortunately quite common. To be fair Sanskrit does have a very precise grammar and sentence structure, and a very (very) vast vocabulary. Similar to Greek and Latin, it was the language of science and literature and the elite, it was the language that one wrote books in. Many Sanskrit plays feature high-born men talking in Sanskrit while women and low-born men spoke Prakrit. So it was seen as a superior language, perfect when compared compared to Prakrit and other "common" languages.

This is my first time answering on this subreddit, I hope that I followed all the rules. Also I'm happy to answer any follow-up questions!

I don't have an answer, but do have something you might find intresting.

About Muslims and translation part. Well, they only mean it for the Quran, in the sense that every translation will always be a translation and not The Quran. That's simply because not every word has a perfect equivalent in another language. And Quran formed basis of law in many Muslim states for centuries, so it was better to go for the exact Arabic word when presenting your arguments instead of using a translation.

However, one thing of interest is something I've heard in Pakistan (might be in other cultures, can't say). Now people in Pakistan converted from Hinduism, and mostly through Sufis rather than more formal self study or anything.

And as it is, some suras (chapters) of the Quran start with letters that don't form any word. From what I've heard, this practice was common in speakers of classical Arabic, but then died out in later years so meaning of these letters was lost (maybe you could guess it by comparing it with poems in classical Arabic employing the same thing, but you can't know the exact meaning since this is something no longer used so we just can't be sure). Now, I've heard some people from Pakistan say if someone can understand meaning of these letters, they would gain some kind of "super natural powers" (they didn't explain what though). Now early Muslims - who were speakers of classical Arabic - defiantly understood meaning of these letters. Yet they din't have any super powers. And yet this "myth" or whatever is present (though remember, most people in Pakistan live in rural areas. And most of their religious knowledge seems to come from their community rather than through study).