First the obvious: the word offends those who think the word puts us in peril of hellfire and brimstone (although it can be argued that the f-word is a more accurate translation of scriptures like ). Many consider it more acceptable to go out of one's way to offend a religious minority than other minorities, such as by using the n-word. But in your case, perhaps you are an equal opportunity offender (have any Turks objected to your occasional use of "Turkish butt-stabbing"?) So let's consider how the word is used in a context where offending religion is taken for granted. 

 In addition to the word's anatomical sense and its intensifying sense, there is also an anti-intellectual connotation. So it's hard to imagine how it would be used constructively to discuss something like the Pythagorean theorem. An anti-intellectual connotation is occasionally helpful when something has been analyzed in great detail while missing something obvious. However, if you just want to say that the Emperor has no clothes, then any informal English will achieve a similar goal. 

 Another connotation of the word is that it can often be used as fighting words. In "Why the fuck did you even get involved?", the word changes a relatively civil question into an invitation to a brawl. So some of your examples like "Fuck Wally" strike me as uncivil. "Disregarding Wally" means the same thing without unnecessarily emphasizing a primeval challenge.  

 The intransitive example should be "Wally and Thorn fucked", not "Thorn was fucked by Wally". That passive construction requires a transitive verb. If I substitute intransitive verbs, the sentence becomes obviously ungrammatical: "Thorn was farted by Wally." "Thorn was peed by Wally." Perhaps what was meant is that Thorn was farted at and peed on by Wally, but that sounds OK only because Thorn has been changed from the ungrammatical object of an intransitive verb, to the wet object of a preposition.