Interesting examples.
on the evenings
between 6 and 9 on the evenings
refers to the "evenings" on the days you are working, whereas
in the evenings
between 6 and 9 in the evenings
refers to the time during any evening (without additional context).
You could rephrase this as
On the days that I am working, I am free between 6 and 9 in the evening.
In my opinion, yes, "to go fencing" has a right to exist. Note that this is an opinion, just like everything else about what is correct or not, and this is the kind of opinion that fluent speakers reasonably disagree about. What I can do to help you form your own opinion, though, is tell you my reasons.
"Go ____ing" is usually used for activities that require travel to a different place to do them. The reason you "go" fishing is because you can't do it here. You have to go to a pond or a creek or someplace like that. If you danced at home, we don't say that you "went dancing"; you just "danced". We say "go dancing" to mean that you went someplace, away from home, like a dance club, and danced there.
"Go ____ing" also suggests that the activity is being done by amateurs, not professionally (also not by very serious amateurs of the sort who might go to the Olympics). Professional fishermen, for example, are usually said to "fish", not to "go fishing". A ballet dancer is said to have "danced at the Royal Opera House in London", not to have "gone dancing" there.
None of that is a rule, of course. English works more by precedents, analogies, and conflicting pressures than by rules. People don't usually say "go boxing" even though usually you go someplace special to box (a gym with a boxing ring) and even though boxing is mainly done by amateurs. However, when the notion of traveling there is salient, people do say "go boxing". For example, this Quora question asks "Is it OK to go boxing after I lift?" This suggests that the speaker lifts weights someplace different than where he or she boxes. Similarly when people speak of "going boxing" as a date, since a date suggests travel to a public place.
Fencing is normally done in a special place that you have to travel to, so it's reasonable to speak of "going fencing"—unless you're already at the fencing club. So, especially if you want to suggest that you are fencing as an amateur, not very seriously, but you do go to a fencing club (rather than just fencing with friends at home), "to go fencing" is reasonable. If you search Google, you'll find plenty of usage of "go fencing".
I think there are two main reasons why some fluent speakers might shy away from "to go fencing". (1) They have never done it and they don't know anyone who has, so they haven't heard the expression "go fencing". (2) Fencing is usually understood as a serious and dignified sport, comparable to ballet. So, "go fencing" might seem to clash with that.
If you want to avoid the connotations of "go fencing", then the best verb is "fence". I don't recommend "do fencing", because it sounds ungainly, but it's certainly not incorrect grammar.
Otherwise, my advice is …
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Interestingly, you can say "a bible" if you're referring to one of many physical, printed versions of the book.
When you say "the Bible," however, you're referring to the literary work itself, rather than a specific instance or printed version of the work. I think we simply think of it as a proper noun, as though "the" is part of the title.
Looking into this I did find this discussion, which may be of interest. It didn't seem to come to a consensus as to why this is grammatically, but some users mentioned the etymology of the word as likely contributing to this usage. "Bible" comes Latin biblia meaning simply "book." Essentially, people used to simply refer to it as "the book," and I suppose the definite article just stuck around while "Bible" evolved into a word of its own.
Worth noting is that there are a lot of historical literary works whose names use the, including religious texts (The Torah, The Qur'an) and epic poems (The Iliad, The Aeneid).