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Don't say anything about her behavior--when all is said and done, her behavior is her business, not yours. However, your REACTION to her behavior is yours, not hers. You cannot change someone else; you can only change yourself. In your case, you know the consequences of your past reactions. Consider reacting differently--it will throw her for a loop, and her behavior will be different. I cannot guarantee that it will be better, only that it will be different, and that will open opportunities for you to redefine the situation and the relationship.
A couple more comments. She is your elder sister by many years according to one of your earlier posts. Despite the fact that you are now both adults, she sees your comments about her behavior as disrespectful and impertinent. It does not matter whether they are or are not--that is the way she views them. Quite frankly, though, so do I. I do find her behavior boorish. If you don;t talk--trying to appease or correct her, then she is the one left standing (or sitting, as it were) by herself a shining beacon of bad behavior for everyone to see. When you respond, you join her, and she has someone with whom to share the blame.
As for the throwing the small dog through the screen door, and "carry your own stuff!" there certainly must be more to the story, but from what you said, those responses were beyond overkill--they were downright rude and mean-spirited. A gentleman OFFERS to help a woman, which included taking and carrying the dog at the airport before you had to be asked. With the dog chasing the cat, first you politely ask her to restrain her dog, if there is no response, THEN you take her dog and put him back into the carrier and say that the dog isn't allowed out of the carrier except when on a leash and being monitored for the remainder of the visit.