Both the -ic and -ous suffixes indicate adjectives. But I'm
wondering what determines the suffix used? So for example, we
have exothermic and endothermic, and we have exogenous and
endogenous. Does the -ic come from the German -ich, the
french -ique, or both? And whence -ous? And finally, and what
initially started me questioning this, is there any rhyme or
reason behind which ending a root ended up with?
I think -ic could come either from Greek -iko- (e.g. idiotic) or Latin -ic- (e.g. civic, where the suffix is applied to a native Latin root that was not borrowed by the Romans from Greek), and -ous would always be from Latin -os-, e.g. perilous from pericul-os-us; my Ancient Greek is not good enough to say for sure that there was no similar suffix in that language, but I can't think of such an example. French -ique is like English -ic and has the same origins as just listed. German hasn't got a suffix -ich (it has -ig, and -lich); but in any case with stems from the classical languages English would normally use classical suffixes, even if the combination was invented in modern times rather than already existing in Latin or Greek. (Until recently, the kind of people in Europe and the English-speaking world who were socially "licensed" to coin long new words for new ideas would all have studied Latin and many of them would have studied Greek, and in drawing on elements from those languages to create novel terms in English or other modern languages they would have aimed to fit the elements together in ways that _could_ have occurred in the classical languages even if the particular combinations happened not to occur.)
This gives you part of the answer to your question how one or the other suffix was chosen (and of course there are many other suffixes in "competition" with these two). Often, the word including suffix will already have existed in one of the classical languages, and (if I am right in thinking that Greek had no -os- suffix) that means that there was no chance that that language could have had a word which would have turned into English "idiotous". Sometimes the classical languages selected between alternative suffixes in terms of euphony: for instance, another adjective-forming suffix in Latin was -al-, as in annual or mental, but with the root famili- they didn't want two L sounds in a row so the alternative suffix -ar- was used to give the word which has come into English as familiar. (Yes, I know there is also a word "familial", but that is a recent coinage by someone who didn't share the Romans' ideas about euphony!) This might explain why "dangerous" is 'perilous' rather than 'perilic'; the root for "danger" in Latin was 'pericul-', so using the -ic- suffix would have given two ic's in one word, which might have sounded silly to the Romans.
But there is a lot more to it than that. Within the classical languages, alternative suffixes had connotations of their own, which would have been appropriate or inappropriate for particular shades of meaning. Compare the way that with native English suffixes, say -ish and -ly can both be used to change nouns into adjectives, but I think many people would agree that -ish tends to have a slightly mocking or downputting flavour, whereas -ly is if anything the opposite. Words like "sheepish", "wimpish" sound natural, and so does "kingly" in another way, but "sheeply", "wimply", or "kingish" would clash because the connotations of the suffixes would contradict those of the roots. It would be a large subject to go into the precise connotations of the various classical suffixes found in English vocabulary; I would not be qualified to do that and anyway a brief Ask-a-Linguist response would not be the place, but the reference books mentions by my fellow advisers would be good places to start.
Hope this helps,