When were the first university or secondary school history courses?
Which was the first English public (i.e. private secondary) school
to teach history and when?  Was history first taught in American or
English universities?  When were the first courses, and when the first
professors specifically of history?

Well, if you believe in the Judeo-Christian version of everything,
then the first history lesson on Earth would have been given to Eve by
Adam.  Since he was the first human, he would have explained to her
how he came to be.  : )

bugbear,

I believe your question is one of the great imponderables...

Imponderables
http://www.imponderables.com/index.php

My reasoning is that in evolutionary thinking, there is no beginning,
or ending --only progression.  In the scope of your question, even
prior to the very first school, the "history" of our ancestors, and
what is known [beliefs] about our existence and the universe we live
in, has always been handed down from one generation to the next.

~~Cynthia

I spent several hours on this, since I found the question quite
interesting. I didn't find much for you, but this list of books might
be helpful:

University of Durham: The History of History Teaching Collection
http://www.dur.ac.uk/g.r.batho/archive/hoht.htm