Etiquette with regard to email consolidation
Do universities/departments ever a formal policy in place about the use of "non-official" email accounts?


Yes, at least in Europe most universities seem to at least frown upon using gmail etc. in professional contexts. Whether there is a formal policy against it will differ from place to place. I would not even be able to tell in my current university, as the thought has not really crossed my mind so far. However, our department head is occasionally reminding people informally that they are expected to be using their official mail addresses.

That being said, whether universities with a formal policy exist isn't overly important for you. You should primarily care whether your university has such a policy, and this we cannot answer for you.


  Do people find it personally bothersome when colleagues don't use their university account for professional correspondence?


Yes, I personally find it somewhat unprofessional, although I have no particularly strong feelings on this. I do occasionally wonder why people who clearly have an official account of my university specifically choose to send emails to collaborators and students via an account like cookiemonster93@gmail.com. It can't be that they prefer the gmail user experience, because, as you say yourself, it is very easy to use gmail also with non-gmail accounts.

However, note that even using the gmail web client as a frontend for your regular university accounts may already be a breach of policy. At least here in Switzerland, universities are not happy about the mailings of their employees being uploaded to Google (but then again they certainly wouldn't use Windows Live either, so maybe your universities care less about data privacy than the ones over here).


  Is there anything I absolutely should not send via a "non-official" account? (There's a good chance FERPA could be involved here...)


I'm not in the US, so I cannot comment on FERPA, but in my university basically everything that has to do with students, assignments, tests, or grades needs to be sent encrypted with the certificate that is bound to our university identity. This also automatically excludes sending it over a different mail address. Every "official" communication needs to at least be sent digitally signed using said certificate, which again excludes other mail accounts.