On the Mr Knightley telling Emma off thing, I honestly think that whole passage is one of the most brilliant pieces of prose description of the many and varied flaws in human nature in the english language. The point is that Emma isn't inherently wicked, evil or nasty, just a normal human being under a certain amount of emotional strain, being irritated by an annoying but well-meaning person and so is betrayed into a objectively unjustifiable and really quite hurtful breach of manners. As someone said, particularly unpleasant for Miss Bates because of the social dynamic that Emma has all the privilege of looks, wealth, power, youth etc and she is the 'poor relation'.
The brilliance that austen subtly draws out though is that while Emma does know in her heart that she was wrong, she pretty quickly performs the sort of mental gymnastics that all we (if we're being honest with ourselves) do when we know we're in the wrong, 'oh it wasn't that bad', 'she didn't even notice', 'she'll have forgotten about it soon', 'it would have made it more awkward to say something at the time', 'best to just forget about it' and by the time she's got one foot in the carriage door she was well on the way to being totally wrapped up in herself and her own concerns again - it takes the shock of someone you truly trust and respect (like Emma does Knightley) to point out to you how wrong you were to shock you into a proper sense of it - also takes a lot of moral upstanding-ness and bravery to challenge someone you love when they've been wrong as well and to do the right thing over the easy thing - the fact that Mr Knightley doesn't quite manage it without coming across as pompous and patronising shows he's not perfect either (and I do agree the whole 'he's been teaching her right from wrong since she was in the cradle' is a bit of a creepy dynamic.
Frank Churchill would never have bothered to say anything to Emma even as part of a joke, because he's all for doing the easiest/most pleasant thing for him rather than the right thing by other people over the whole book, he's set up as a foil to Knightley in that way, and even had he noticed Miss Bates was hurt (unlikely!) he wouldn't have risk causing a row with Emma about it - again though I do think that's the brilliance of Austen that despite him being the 'baddy' of the piece she gives him a proper happy ending too, arguably a happier/better ending than Knightley and Emma get, they get a quiet wedding then living a life of virtue tending to their apple orchards and indulging a querulous old man in their secluded/boring country village where their greatest triumph/excitement is lording it over the rustics, whereas Frank and Jane get to travel the world, wear sparkly jewels and doubtless host glamourous balls in town as well as having a huge estate at Enscombe etc. So while the 'message' of Emma is ostenisbly that truth and righteousness gets it's due reward in the end she also has a bit of a tongue in cheek about the realities of that too...