> Does the compose key work on different keyboard layouts?
Yup, switching to Dvorak and the sequence is the same (. then [ on qwerty). There also seems to be an ibus mode for it in ibus-table-others, along with a mode that tries to map LaTeX to unicode symbols, but I haven't tried either.
As for your other remark, I remember back in the day using the Gaim-LaTeX plugin to communicate LaTeX with people over MSN/AIM/IRC/etc... If you had the plugin, it would auto-render LaTeX for you. I also think there may have been a plugin that would send the other party an image instead so you could still share if they didn't have it. I haven't used anything like it in a long time as my need to communicate or have communicated formatted math to people over IM is pretty much 0 these days, but perhaps https://sourceforge.net/projects/pidgin-latex/ still works. But also on IRC, many rooms have bots that accept programs to eval and print, typically used for math calculations, and then you just use the syntax of the language instead of LaTeX.
For Skype, Telegram, Discord, comment hosts like HN or Discus or blog software, that's a failing on them for not implementing a very well-known standard (TeX)...
But you can still share raw TeX strings, especially since almost everyone at least implements some sort of blockquote or monotext formatting system where it's possible to bypass markdown et al. rules turning 345 into nonsense. And while you may have found it difficult to read the equation I posted earlier, I'd be really surprised if it took you a long time to get comfortable enough to read it and other arbitrary equations around whatever level of math you have in their raw TeX form (especially if the poster bothered to format it a bit more nicely with whitespace, which I didn't do). It's kind of like getting used to reading regexes, but even easier.
I guess maybe you could try to convince me it's more difficult to pick up (even for a high schooler) than I think, perhaps since I've just been used to it for so long and the last time I saw someone pick it up was in college? And of course this is only talking about writing math with the purpose of displaying it nicely in the end, actually calculating will tend to involve another syntax yet again but I don't think those are typically bad either... Perhaps raw Python can be terrible if you're trying to do symbolic math, but then why wouldn't you be using a system explicitly for that or even in Python use SymPy (and I'm just now remembering there's that whole Jupyter ecosystem that I'm sure has support for SymPy and rendering nice looking math with LaTeX). I've enjoyed Maxima, which even in a terminal does a good job with ASCII graphics http://maxima.sourceforge.net/i/maximacl.png but of course there's a graphical front end.
I got a bit off topic but as a last resort to communicating math over such channels, you can go to sites like https://www.codecogs.com/latex/eqneditor.php?lang=en-en and write out your math (it even has some buttons to help if you don't yet know the TeX-isms) and share a link to the output like https://latex.codecogs.com/png.download?%5Cdpi%7B120%7D%20%5... Slack even downloads and displays it for you in-chat.
(It's also worth pointing out that Discord has an ok-ish Pidgin connector... and even my ghetto home-grown blog has had LaTeX support for posts and comments through a JS library that I haven't bothered to update since 2012. It even lets you right-click to view the TeX source too, example near the bottom here https://www.thejach.com/view/2011/9/playing_with_morton_numb... So never say never!)