The best answer that I can give is to adapt the notes I made years ago during instruction in prose composition [which I am doing by inserting comments in square brackets].
ADVERBIAL CLAUSES — CAUSAL
The word cum has a whole list of meanings, as conjunction, preposition and adverb. The form quum is often found instead of cum, and these two are more or less interchangeable as causal particles — that is, to introduce causal clauses — both being found throughout classical texts. The form seen is often that preferred by the redactor [which may be an obstacle when a corpus is consulted to deduce a trend, or a rule based on frequency].
[A little contrary to what the question says] The verb following cum is normally in the subjunctive : but there are a few exceptions — the indicative can be used when the action of the verb is future or present (e.g. after a few 'verbs of emotion' such as laudo, doleo, gaudeo). Note that in adverbial clauses that are NOT causal but temporal, introduced by cum = 'when', the indicative is legitimately used in various situations.
[For each of the words in question] English from Latin is chosen to suit the style of the translator: since, as, because, seeing that, for the reason that etc. For doing into Latin, on the other hand, Quia and quod can, like cum, be used to translate any of these and, like cum, are used with the subjunctive — with indicative ONLY if the reason is given with certainty:
advenerunt quod cibum susciperent but advenerunt quod cibo egebant
- Since causal clauses are syntactically adverbial, the introductory particle can refer to a reason shown in the main clause:
cibo egebant, idcirco advenerunt quia cibum susciperent
4.To conclude : When used as logical conjunctions, the differences between cum, quod and quia are minimal. The latter is most often used by the subject of the principal clause in expressing an opinion and is therefore considered rather more emphatic : such usage is not obligatory. The differences hardly show in a good, idiomatic translation from Latin to English : such difficulty as occurs lies in the opposite case, of choosing the most suitable word of Latin to suit an English original. Here, too, teasing out the niceties of difference is a somewhat arid pursuit, for there may be many ways of saying the same thing in English on these occasions, just as in Latin : in the end, it comes down to personal style and preference. In that connection we should remember that not only clarity, but also elegance is desirable in a Latin prose that aspires to high quality.