If Henry Higgins is a linguist, and specifically studies accents, why does he have a whole song where he rants about how everyone should speak RP? Because if everyone spoke the same, there wouldn't be as much for him to study, and you'd think he wouldn't be in a profession where he purposely listens to speech he dislikes. I guess one explanation could be that he likes being a linguist but he still gets annoyed at hearing accents he doesn't find appealing. He still pretty much insulted everything but RP in that song, though.
If you want to fix something, first find out what's broken. Perhaps Higgins believes that in order to fix the problem of people not speaking properly he needs to study all of the differences in great detail.
Many early 20th century linguists still worked under the misconception of prescriptive linguistics, that they should find out what "true" language should be, and educate it to the masses, as opposed to observe normal, everyday language, and determine what rules govern its use without outside interference.
Also, he's arrogant. Just because he studies accents doesn't mean he can't think one is inherently superior to others. Just the same way a person who studies insects might have a favorite.
He observed that less-than-RP language was strongly tied to social stigma and tightly associated with the lower classes. He asked himself: "simply by talking in more refined ways these people would shed the obvious tie to their background and make improving their life easier; why don't they do it?"
Why is Eliza so angry at Higgins after the ball? He hasn't acknowledged her merit at passing as a princess, true, and he hasn't been that nice to her. But both Higgins and Pickering have given Eliza a luxurious life for six months, have taught her refined speech and good behaviour, have introduced her into high society, and most important, have given her the chance for a better future, either by marrying into a rich family or starting her own business and remaining independant. Otherwise she would have been in poverty all her life. I think there's more to be thankful for than to be vengeful for.
She's certainly appreciative of that, but hearing Higgins and Pickering congratulate each other shows her that they pulled her out of poverty only to settle a bet and didn't actually care about her well-being. She liked being with them and now thinks they're just going to push her back onto the streets now that they don't need her anymore.
Exactly. They're patting each other on the back, and acting like her hard work, development, investment, sacrifices, etc. etc., were nothing. They weren't even acknowledging her a little.