Here we are again on the intricate, "confusing" network that foreign speakers need to navigate around the "right", "wrong", "maybe right, maybe wrong" way of addressing and being addressed when around Chinese people, whether in China or elsewhere.
First of all there isn't a one-size-fits-all words or phrases that cover all or any delicate or indelicate situations, social or private, like "Miss", "Lady", "Madam" which are ultra safe in 99% of situations in English speaking places.
To answer your queries:-
(1) Parents in China routinely introduce me to their children as 爷爷.
Family relationships are of the utmost importance to the Chinese and introducing you as "grandfather" to their children, (who actually know you have no blood relationship to them), is a mark of great honor and respect not just for you as a person but also you are now officially part of the family, (whether you like it or not), and for your old age as well which is historically much treasured.
(2) The manager came over to my table and said 姐姐给你看.
She said "Let elder sister show you how its done", (though she could be 40 years younger than you) Now, depending on her tone of voice and whether it was with a smile, it could, one, be just a straight forward professional gesture to help a paying customer, or, two, as the " 姐姐" part might indicate, an act, (bordering on playful sarcasm), of an elder sister instructing her younger brother how to do some, to her, really simple tasks, something which an elder sister should teach a younger brother.
She might be factually wrong in terms of age-relations between you two, but she was right from the pedagogical point of view. But then again, she could just be down right sarcastic towards an old clumsy foreigner.
(3) Could she have called me 叔叔 or would that be wrong somehow?
No, not wrong, as the Chinese have now taken to using "Uncle" rather loosely to mean a male of your father's age, like all things "Western". However, we have “USD” which does not mean US Dollars but 糖叔叔爸爸, i.e. Uncle Sugar Daddy, (USD)
(4) Could she have called herself 妹妹
Well, to the Chinese, especially the older generations, knowledge, wisdom are in the domain of the old and aged. She has to be your 姐姐 to claim the "right" to teach you anything. In certain playful situations when a ten year old kid tries to be argumentative, the adults may condescendingly say, 是，你老人家说的对。