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