Your memory is not far off. Mark used failed logins to access other people’s emails:
https://www.businessinsider.com/how-mark-zuckerberg-hacked-into-the-harvard-crimson-2010-3
I cant say if he intentionally saved passwords unencrypted, but he certainly saved login attempts
I have a domain with one of the new TLD which I used for my emails.
Most services worked fine with it, but there were a few cases where my email was flagged as fraudulent and I had to call, explain it was legit and provide with another email.
There was one service I registered which explicitly said they oy accept gmail addresses.
Roughly one year ago I acquired a new domain using the .org extension, I am migrating my accounts to this one, and I havent had any problems so far.
So overall my conclusion is that most services are fine with custom emails, a few of them block based on TLD and an even smaller subset will allow only specific providers. Since I am moving alway from big corp, having a widely used TLD that seems to be accepted in most cases is my personal sweet spot.