Harvard was the first American institute of higher learning, and Nathaniel Eaton it's first headmaster. Harvard was a college at this time, not a university, and Eaton the only teacher. He was also fired fairly quickly, and replaced with Henry Dunster, a Cambridge graduate who stayed longer and established Harvard as a real center of education. Many would consider Dunster the first true American professor, again of a college, not a university. Here is a quote from Samuel Eliot Morison's Founding of Harvard College:

"Dunster found Harvard College deserted by students, devoid of buildings, wanting income or endowment, and unprovided with government or statutes. He left it a flourishing university college of the arts, provided with several buildings and a settled though insufficient income, governed under the Charter of 1650 by a body of fellows and officers whose duties were regulated by statute. The Harvard College created under his presidency and largely through his efforts endured in all essential features until the nineteenth century, and in some respects has persisted in the great university of today."

The first named professorship in the US was the Hollis Professorship of Divinity at Harvard. Edward Wigglesworth was the first person to hold this chair.

As far as the first school to claim the title university instead of college, that one is a little more open to interpretation and the schools in question had multiple professors when they grew enough in scope to be considered a university.

This depends on some semantics. You mean first after the founding of the US, or first in US territory? And first at any institution of higher learning, or only at a self-styled university? And any college/university level instructor, or someone with a chair?