What is the reason/background of the "des" in the phrase des Weiteren? Why is the genitive case used here?
I would say it is an adverbial genitive. In most cases the genitive is difficult to explain. I haven't done any research on this topic but there should be some information on the internet as it is an interesting topic.
Such old forms are dictionary entries and for each such entry I would consult the online dictionary DWDS which has etymological information and a lot of example material.
DWDS has registered "des Weiteren" in the entry weiter, but they have no comment on the genitive form.
The answer is quite simple, if I am guessing correctly. It may be akin to "fernab dessen" (besides), "Dies liegt fernab dessen" (this lies far apart from that) reposed as "dessen fernab", with elipses "des' fernab". An analogy could be seen in "abgesehen davon", "davon abgesehen, davon ab".
Compare especially, "deshalb" - which means ca. "on behalf of, due to"
Weshalb ist das so? Weil das so ist, deshalb! Deshalb ist das so.
Why? because that's how it is, thats why. That's how it is, due to that being that.
We know wegen dessen, which fits into the same pattern as wegen ihm, which is clearly dativ, but "wegen seiner", genitiv, is not unheard of either, especially in "wegen meiner ..." (for all I care) which I personally understand as variant of "meinetwegen", "wegen mir", and as elipsis "wegen meiner [Person]".
-halb- is an opaque morpheme in German, appearing also in allenthalber and perhaps the same as in Halbzeug, though that's traditionally explained as half-finished. halb "half", Hälfte, etc. are unexplained etymologically. I don't know whether * "halb dessen" did exist or would make sense, and needed genitive inflection (or rather participle * "halbend dessen"). In the end, I am still just guessing. Compare also, "aus diesem/demselben Grund ist das so".
This would be the model the other cases, that you are asking for, are built on, if learning the language by ear. First a fixed lexeme is recognized on the morpho-syntactic level, and later as a pattern is lexicalized, merging with the usual genitive inflection.
So einfach ist das.