Can you give an example? Because I have no idea what you are talking about.

The song that you probably have in mind is the tune of “The Streets of Cairo or The Poor Little Country Maid” which was written by Sol Bloom for the Chicago World’s Fair in 1893. I wrote about the context of its composition and the advent of Belly Dancing in a post here: https://amp.reddit.com/r/AskHistorians/comments/4o2243/tuesday_trivia_forgotten_cultural_fads_and/d492g0f/?context=3

In other words, if this is the tune you have in mind, it is a modern, Western fabrication, but a fun one with an interesting back story.

I don't know if this answer will be acceptable, because I don't have sources for everything at hand anymore and it was a long time since I studied this topic, but as a former music student, I can hopefully tell you at least some things. I can't answer the first question, but maybe I can get you closer to having an answer to the others.

Generally speaking, in order to reconstruct music from ancient times, there are a few things that we need to know:

1. Instrumentation: What actual instruments were used? Can we make comparable instruments still, or have some important details been lost? Etc.
2. Musical scales/temperament: What notes were the instruments tuned to play? The thing here is that we are mostly accustomed to what's called the equal tempered scale, wherein 12 notes are equally distributed according to steps of the 12th root of 2 (https://en.wikipedia.org/wiki/Equal_temperament), which is itself a compromise solution with some interesting reasons behind it... But the important thing here is that the scales we're used to hearing and playing our instruments with did not exist, and probably could not have existed. So what notes did they use? What was their tuning basis? How precise were their tuning and construction methods?
3. Compositions: Basically, how were the instruments played to make coherent pieces of music? Is there a written record of some kind, and if so, how well are we able to interpret it? Is there a surviving tradition that we have good reason to think represents ancient compositions?

There might be some more things we can think of (space and acoustics, cultural context, etc), but the list above should get us basically to the heart of the question of the music itself. As a general note, I should say that it is completely inevitable that all of these elements would have changed over the incredibly long lifespan of ancient Egypt, so probably any meaningful historical accuracy would have to try to narrow down a timeframe.

With respect to instruments, I'm aware that we have written and archeological evidence of the types of instruments they used. There are also academic efforts also to reconstruct these instruments (https://web.archive.org/web/20050531222650/http://www.phmusic.gov.eg/news/Natinal%20project%20in%20english%20.doc). Most of the instruments they made are no longer made, though basic ideas are familiar in many cases (as with flutes and stringed instruments). Still, I should stress that even the reconstruction of their instruments is an ongoing effort. So in order to have a chance at authenticity, we would probably need a composition that only requires instruments that we can accurately recreate with some degree of confidence.

With respect to scales and temperaments, we can get some ideas from existing flutes. Flutes are good because if they're well-preserved, we might even still be able to play them! They give very good ideas of notes and note ratios. An example of this type of investigation can be found here: http://egyptsound.free.fr/fathi.htm (Note that you can hear an actual ancient Egyptian flute being played from that page, e.g. http://egyptsound.free.fr/sounds/EgyptSound_RealAudio/01_Flute_69817.ra ) With this information, we can indeed reconstruct at least some of the scales they used, and we can also conclude that some of those scales remained consistent across time and geography, which gives reasonable confidence that by using one of those scales, we would have a meaningful claim to be using the scales of (at least some extended period of) ancient Egypt.

The question of surviving compositions, however, seems to be where we fall short. Basically, we have attempts at "creative reconstruction" (https://www.amazon.com/Ankh-Sound-Ancient-MICHAEL-ATHERTON/dp/B00000AEP0) that are about as good as it's likely to get. Even if we were to accurately identify a written record as some kind of musical notation (which, to my knowledge, we have not when it comes to ancient Egypt), the interpretation of that notation would still be exceedingly difficult. To give a rough idea, our current musical notation system revolves around implicit trained knowledge of what each position on a musical staff represents. We know that a certain note is a "C" or an "E" or whatever because we decided on these values somewhat arbitrarily, and there's nothing inherent to our notation system that might tip a future historian off about which note is which if they only had access to a musical staff and the extra-notational knowledge had been lost. Even when we have some surviving ancient musical notation (as is the case with ancient Mesopotamia), attempts to realize it have been somewhat varied because they are all still "interpretations".

The popularized sound you're used to associating with ancient Egypt is unlikely to be derived from any meaningful historical evidence, though the cultural influence of ancient Egypt may mean that some of its musical features have survived in various forms in musical styles of the modern middle east. In any case, we have no direct surviving music from them, so various interpretations and guesses are about as good as we're getting.

As to how we got the ideas we have and when, I'll have to leave that to others.