There is no universally accepted rule about single/double quotes in BrE. It depends on the publisher's style. British books usually use single quotation marks, but British newspapers and magazines usually use double quotation marks. In everyday use, British people use whichever they feel like, usually fairly consistently (though some favour single quotes for quoting words or idioms, double quotes for quoting direct speech).
(As you know, single quotes are always used for a quote-within-a-quote if double quotes are used for the outer quote, while double quotes are used for a quote-within-a-quote if single quotes are used for the outer quote.)
American style may have some variation too, but it is true that American publishers usually favour double quotes and putting full stops inside quotation marks. (British publications usually put full stops instead quotes only if the full stop can be regarded as the end both of the sentence and of the direct quote, e.g. Jim said, "I'm hungry.")
So when you see this:
He called this phenomenon "the memory of water".
This is likely to be British.
British and American style are the same with respect to question marks. In other words, if the question mark indicates that the quoted material is a question, it comes within the quotes:
He asked, "Have you ever been to France?"
But if the question mark has nothing to do with the quote and marks the sentence as a whole as being a question, it comes outside the quote.
Did he call this phenomenon "the memory of water"?
As far as I know, no major American style guide disagrees with this.