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
"How it turns out" is also often phrased in the form of, "tell me how it went". "turn" and "went" are directly related, as "went" comes from an old word "wend", which means "turn".
Isn't that interesting? When you ask how something went, you are literally asking how something "turned" out.
Went is the past tense of go. Turn represents just that, rotation or revolution, a thing going.
Best Answer
Well, the phrase was older than I expected. NGrams reports the following matches for the phrases "got another thing coming" and "have another thing coming":
NGram image http://ngrams.googlelabs.com/chart?content=got%20another%20thing%20coming%2Chave%20another%20thing%20coming&corpus=0&smoothing=3&year_start=1880&year_end=2000
Here is quote from 1906 (I think. I am still learning how Google Books works.):
As for your other question, "you've got another think coming" wasn't an established idiom as far as I was aware, but the NGrams results shocked me:
The usage and meaning seem identical to "thing" but I find it odd that I don't recall ever seeing it in print. Looking to phrases.org.uk:
Most of the other sites I checked said similar things. Namely, "think" is the correct version and "thing" is the malformation. The origin appears to revolve around someone thinking one thing but an apparent correct will be coming shortly: They will need to rethink their previous thought (and ideally arrive at the correct position this time.)