Am I a fee basis government official if I'm employed by a state university hospital?

No, a fee-based public official is an individual who receives and retains remuneration directly from the public. 

An individual who receives payment for services from government funds in the form of a wage or salary is not a fee-based public official, even if the compensation is called a fee.

For more information, you can reference the Social Security Administration page below:

https://secure.ssa.gov/poms.nsf/lnx/1960001625

What about a public high school coach who, at the end of the season, gets paid a one-time stipend?

No the stipend is added to your W-2 wages and is does not make you a fee-based public official.

 

 

Link to more information about Fee Based Public Officials

 

 

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