What is actually confidential in communications in state universities? I have been hearing things like nothing you write in an email is confidential in state universities because of the Freedom of Information Act (FOIA) or other similar state laws.

Questions:


Can I file FOIA request to see all the emails that a professor receives including all of their unpublished research progress and interactions with their students and postdocs?
Can a postdoc request to see the recommendation letters received or sent by their advisors about them?
As Nate notes, it is challenging to answer a question about fifty different state laws. By a happy coincidence, though, I sort of did it in 2005 :-)

At that point - it may well have changed in the last twelve years - 


Every state had some kind of local freedom-of-information law, and in general they all appeared to affect public universities.
In some states, private universities may be covered to the extent that they are spending public money.
Academic institutions operated by federal government departments (eg the National Defense University) are subject to the federal Act. Public universities in DC (all one of them) are covered by local law; it varies in the overseas territories. And finally, there is a lot of ambiguity around what FOI legislation, if any, applies to the tribal colleges and universities.

Exceptions
Delaware had a strange law that appeared to exempt the universities in general, but the Boards of Trustees, and financial records, were explicitly covered.
Kansas had at least one specific exemption for a specific part of a university (I am sure other states had similar provisions buried somewhere I missed).
Several states had legislation that only gave rights of access to state residents (though this was constitutionally dubious and one had been successfully challenged).
Explicit topic exceptions were often provided in the law, and most frequently tended to mention recruiting for university presidents, library records, examination processes.


That answers the basic question of "can I use FOI for a university": yes, if they're public, with some caveats. This is generally consistent with international practice.

For the more specific questions, the only possible answer is "...it depends massively on local legislation, caselaw, and practice". I would strongly recommend you look at the relevant state's legislation. If the law is well-established and well-used, the university may even have some handy guidance on this.

My gut feeling, generally speaking, and no more than a rough guess, is that:


  
  Can I file FOIA request to see all the emails that a professor receives including all of their unpublished research progress and interactions with their students and postdocs?
  


Many jurisdictions would balk at "give me all of person X's emails" (as opposed to "give me all discussions about Y". It may well be seen as unreasonably complex to provide, if there is any kind of review stage - the costs for having someone read through all the email to check there's nothing that shouldn't be disclosed ramp up fast, and this can sometimes be a get-out clause. ("We would do what you ask, but we'd have to hire someone for three months, so...")

For ongoing research, there is sometimes an explicit exception, or indeed an implicit one, in the law. (For example, some legislation takes the position that if something is going to be published and that process is underway, it's fine not to release it until publication happens - which would implicitly cover most academic research.)

On the interactions with students &amp; postdocs... "interactions" is an odd word. Routine emails between a couple of public employees (which is another way of describing a professor and a postdoc) wouldn't be automatically problematic. But it's almost certain that something in the full range of "interactions" would be considered confidential. For students, I would assume the assumption of confidentiality would be drawn wider than for postdocs - but, again, local law will be your guide here. 


  
  Can a postdoc request to see the recommendation letters received or sent by their advisors about them?
  


Maybe. It is a borderline issue and one likely to be handled differently in different jurisdictions. It is complicated by the fact that one party is involved in the request, and there may be other more appropriate ways of handling this than FOI. There may also be explicit or implied promises of confidentiality for referees that are enough to ensure they are treated as fully confidential.

Note that "received or sent" may be significant - in the UK, for example, which has fairly robust personal-information and FOI legislation that is probably stronger than that in most US states, you can get copies of recommendation letters received by the public body, but not sent by them. 

So on the whole... it varies. At least you know the law probably applies to some degree!