ProtonMail is indeed great and offers a bridge for use with IMAP-compatible programs. Check out https://protonmail.com/bridge/. I believe they are working on a more integrated solution, but no news of when it'll be out have been announced.
Have you tried looking at VIM-LaTeX (http://vim-latex.sourceforge.net)? It does exactly what you described and is just as customisable as auctex.
For example: <leader>ll will compile your document, multiple times if necessary, and show you warnings/errors in a new window, letting you hit enter on each to jump to the line. Lots of helpers for inserting environments, items, packages, too!
But if you're used Emacs just for auctex, more power to you for harnessing the power of two awesome editors!
I do agree that it would be great to be able to use your own mail client on iOS. Not sure that will ever happen though.