starting at the beginning

There is a range of notes, theoretically, that goes infinitely high and infinitely low. Even though this is true, we can not hear a large amount of them because of our limited auditory receptors. Also, there is an infinite amount of notes, theoretically, in between any two notes, or any half step. "How is this possible??" you ask. And then I answer:

Imagine the infinite amount of tones I described as a slide whistle, or a trombone, that is infinitely long. The range will never end, you can always make the note you play higher, or lower. Hypothetical situation: on that slide there will be a place where someone tells you is a C note. Then, they will let you know that to play a C sharp, you must slide up an inch. "If you slide too much, the note will be too sharp, and if you slide not enough, the note will be too flat." they say.

Pertaining to this, I have a question. Aren't those tones, that are between that C and that C sharp, still notes? Why do they not have names, why do we never use them? Why, out of all the ways one could divide an infinite stream of tones, do we divide them into octaves? Why do the note names, as a scale continues upward, repeat after 8 notes? Why, on a piano, are there only 12 note names? Just 12...

Is there actually something, physically, either in the sound wave or the way we perceive the wave, the "same" about a C in one octave and a C in the next octave. Musicians like myself are able to tell when given two notes in different octaves, if they have the same note name or not. How?? Have we just been trained, through years of hearing and playing music based on octaves to think that they sound similar; or do they actually, literally, technically, sound alike?

Imagine if our defined note names were closer together on the infinite slide. Our half step used to be C and C sharp, they were an inch apart on this slide. Now we have new note names--Z and Y--and they are only half an inch from each other on the slide. We used to go from C to C on our piano with 12 notes. Now, how about we only have 10 notes, from Z to Z.

Would anyone be able to make music out of this system, or would it feel so repulsive that we could not bring ourselves to understand or appreciate it. And if were too repulsive, then would this be because of our innate sense of music that we are born with (to think in scales of 8 and certain amounts of space between notes), or because of our learned sense of music from society?