No, a napkin is just a posh word for a serviette. Everyone would be bemused if you said that you used napkins for your periods.
Hogarth’s right - Rowling’s indicating that Ron took some sandwiches from the table before they all disappeared, so that he could eat more later; it’s not the normal way to carry sandwiches around with you.
A tease on her part, and a source of humour and ‘in jokes’ to her friends, which, greatly to her surprise, was taken seriously by the public. Several of her earlier works were republished not long after the U/non-U essay. She was greatly annoyed that, as a result of the essay, she was forced to go through the earlier works and edit out all of the non-U usages from her characters’ speech. It was felt that her readers would not tolerate such linguistic solecisms.