Learn English – Origin of phrase “open-and-shut” as in “it’s not an open-and-shut case”

Here, fast doesn't mean "(capable of) moving quickly". Much rather, it is being used in the sense "firmly fixed" (see fasten your seatbelts or fast friends). The Phrase Finder says that "This is a nautical term. A ship that was hard and fast was simply one that was firmly beached on land." It adds that the term was used in figurative sense by the early 19th century.

Personally, I don't think it's being used more commonly in the negative. Indeed, a quick COCA search returns 40 occurences of "hard and fast rule" or "hard-and-fast rule", but only 22 of them are using it in a negative context — and I am being as generous as possible there, counting not only "no hard and fast rule" and "not a hard-and-fast rule", but also "don't have any hard and fast rule", "rather than any hard and fast rule", "was never a hard-and-fast rule" and the like.

One thing stands out to me, though: out of 16 occurrences of "hard-and-fast rule", with hyphens, 12 appear in a negative context, or 75%. For the non-hyphenated version, it's almost the other way round: 60% positive, 40% negative. (Again, counting "negative" very generously.)

The figures from the BNC are too small to be statistically meaningful. But anyway, here's an overview:

                              COCA                   BNC
                        total   negative      total   negative
hard and fast rule        24       10            8        7
hard-and-fast rule        16       12            1        1 

William Kitchiner M.D. (1775–1827) was an optician, inventor of telescopes, amateur musician and exceptional cook. His name was a household word during the 19th century, and his Cook’s Oracle was a bestseller in England and America.

^Wikipedia

The phrase forty winks, meaning a short nap, can be traced back to Dr. Kitchiner's 1821 self-help guide, The Art of Invigorating and Prolonging Life. The phrase is mentioned in a November 1821 issue of the British Literary Chronicle, in a review of Kitchiner's book:

Sleep is a subject on which our author acknowledges his feelings are tremblingly alive; he is fond of a 'forty-winks' nap in an horizontal posture,' as the best preparative for any extraordinary exertion, either of body or mind.

Here is a clip from an 1822 copy of Kitchiner's book:

The use of quotes around a forty winks nap seems to indicate Kitchiner might have borrowed the phrase from elsewhere, but I can't find it in any form earlier than his use of it. Also, Kitchiner carefully footnotes other phrases and passages from different authors throughout his book.