Since I started this thread, I think I should also contribute.
In my school years
(I am talking about the '70s, it is about time I edit my profile and confess my age
), there was a dissociation between grammar and literature, more or less the way you describe in your posts. There was a subject called « French », but we had separate hours and used different manuals for grammar and literature. I am under the impression that grammar teaching was cut off from its concrete application in speech, literature, newspapers... everything.
By that time, grammar was mostly considered as a tool for correct writing and spelling
(see CHERVEL, A., ...et il fallut apprendre à écrire à tous les petits Français. Histoire de la grammaire scolaire, Paris, Payot, 1977). Remember, no personal computers, no spellcheck, only paper and pen and your writing skills... so one needed to know grammar rules to write correctly when applying for a job, for example. Proper writing (and proper speech) were factors of social integration.
This way of teaching often results in a kind of 'grammar book literalism', analogous to 'Bible literalism' (or any other holy book. I wonder if there is a connection, btw).
I have a rather literal example of the grammar book used as a holy book...
As a child, I didn't like French grammar classes very much, but I enjoyed creating examples on my own to apply grammar rules. I was extremely frustrated in « 4e » (I was 13 years old) when I had an old-fashioned teacher. We had to learn entire chapters of the grammar book by heart, including examples, and we had to recite the whole chapter in writing every week. I got punished for using my own examples instead of quotes by XVIII century writers, and my parents were summoned to school for this reason...
That teacher I had was right in one thing: learning grammar helps develop a logical reasoning about the way you use language. She used to say that was also true for Latin, which is generally considered « more difficult » (thus more stimulating for the brain

) than French because of nominal declensions. But I would say it is true for any language learning. In France, many people believe that nominal declensions are difficult. However... talk about the complexity of verbal tenses in Romance languages!
That lady could have made me turn away from French grammar and literature. Luckily I had other teachers and I read a lot, too. I concur with Etcetera that reading helps much if you have a « photographic » memory of the texts that enables you to reproduce the structures you have learnt. But having an overall knowledge of grammar to analyse linguistic phenomena also helps.
Grammar is taught here, however, the general notions like tenses, grammar, etc. were all taught at a rather early stage of schooling. As school progressed it has become assumed as already known knowledge, which in a way is bad...
This is also true here. French grammar is taught in primary and part of secondary.
Within what was considered "mother tongue" we had two separate subjects, grammar on the one hand, and something along the lines of "creative writing" on the other, which was actually quite a broad topic, including essay writing, debates and discussion, oral presentations etc.
« Creative writing » also disappears from the French secondary system as school progresses, unless teachers wish to adopt this approach (« ateliers d'écriture »), but not all do. The standard writing test or exam is an essay, a dissertation or a text analysis that follows well-defined guidelines. Correctness in grammar and spelling is of course evaluated, but the aim of this exercise is to develop the ability to follow a plan.
A typical French spelling and grammar nightmare is the agreement of past participles. French spelling is quite difficult by itself, and if you add the fact that agreements in gender and number are not always reflected in phonetics... woaw. Spelling is a national sport of sorts, and we have TV shows with dictation contests... The rules about past participles + avoir or pronominal verbs are so confusing that even the most traditional grammar authors (Grevisse...) admit that some speakers or even writers do not apply them all.
Another classical "headache" is tense concordance, because some verb tenses are becoming obsolete and there are many exceptions to the rules. Who said the French had a Cartesian, logical mind?
Oh -- and thanks to all for your great contributions so far!