In "Pygmalion", what opinion does Higgins have of himself and how does he set himself apart from middle-class men and women? Is that why he set ...

Higgins is
quite sure of himself and confident in his own intelligence.  Lets be honest - he's arrogant. 
He lives the way that he wants to with little concern to what society wants.  Here is how Shaw
describes him:

careless about himself and other people, including their
feelings. His manner varies from genial bullying... to stormy petulance... but he is so entirely
frank and void of malice that he remains likeable even in his least reasonable moments."

He is careless in his decision to ignore propriety.  He doesn't adhere to
Victorian principles requiring a certain amound of tact and reserve.  He says what he wants when
he wants to.  In the last act, he is admonished by his mother, who insists that he
"behave" accordingly if he wants to speak to Eliza:

MRS.
HIGGINS: If you promise to behave yourself, Henry, I'll ask her to come down. If
not, go home; for you have taken up quite enough of my time.

HIGGINS: Oh, all right. Very well. Pick: you behave
yourself. Let us put on our best Sunday manners for this creature that we picked out of the mud.
[He flings himself sulkily into the Elizabethan chair].

By not
"behaving" according to society, Higgins sets himself apart.  He arrogantly believes
that he does not need to be anything but himself and everyone else should just accept him. 
Eliza does reject him for this.  All she ever wished for was an average, socially acceptable
life.