I live in the USA.
I was born in England.
My father’s father (my grandfather) was born in France.
My father has a French passport (he was born & raised in England), and when I was a child, I was on his passport (you used to be able to have your kids on your passport also).
Am I considered a son of a French citizen since my father has a French passport (though he wasn’t born in France, and his mother is English), and therefore eligible to get a French passport?
Being born abroad or having a parent with another citizenship are not incompatible with French citizenship. Unlike British citizenship, it's perfectly possible to transmit French citizenship over several generations. One limitation is that you have to make use of it, lest it lapse after 50 years but if your father regularly renewed his passport and you were on it, it should be reasonably easy to establish that.