it's interesting to note (at least to me) that some emoji outside of the skin tone modifiers are already constructed like this internally - the spaceman emoji is encoded internally as a person emoji and a rocket emoji with a special "connector" character placed between them - the rocket emoji "modifies" the person.
https://dev.to/joeattardi/how-emojis-work-3a2g
so I have no idea why there's not already say, a left arrow + the finger gun emoji to create a left-pointing finger gun, outside of the extremely slow application process you have to go through to suggest changes to the emoji system.
eh. emoji are weird.
You're right. According to emojipedia that is the 'unamused' or 'meh' face, while the one that OP reviewed is the 'Neutral face'
Yes you can use the private space for you own, for example the is only in the AppleColorEmoji font, not on other platforms. It shows as an Apple logo in that font.
You can absolutely swap fonts, though some have licence restrictions on them. I made an app that did that on Android years ago, though much better solutions are available now.
I know you already got an answer, but here's the emojipedia page on it: https://emojipedia.org/wind-chime/
Also, if you come across another one you don't know, you can literally copy/paste it into the search field on the front page.