is Higgins very self-aware? How do you know?

Being self aware, Higgins would have to know himself as Shaw so aptly puts him,

"His manner varies from genial bullying when he is in good humour to stormy petulance when anything goes wrong; but he is so entirely frank and void of malice that he remains likeable even in his least reasonable moments."

So does Higgins know he's a whiny moma's boy? I think to an extent. He recognizes he is rather a jerk,

"The great secret, Eliza, is not having bad manners or good manners or any other particular sort of manners, but having the same manner for all human souls: in short, behaving as if you were in Heaven, where there are no third-class carriages, and one soul is as good as another."

The problem is weather he is sincere about anything he says. From this point I would say he isn't self aware enough. Higgins tends to justify his actions as if he's entitled to them. He isn't entitled to them just because of being borne fortunate and he does't get this.