I think the better verb for what you're asking is "signal" rather than "determine" – the formal marking of grammatical function is caused by a word having a particular grammatical function, not the other way around.
Second, you seem to be specifically interested in case-like functions a.k.a. grammatical relations – there are many kinds of grammatical functions.
The first thing you should also allow for is that in some instance, a particular grammatical relation is not marked at all. Or, you might say that the marker is "null". This is the situation with your Turkish examples and in Finnish as well, that subjects do not have a case marker. Turkish and Finnish do have case markers, which are in fact suffixes, which is to say, things added after the stem within the word, to mark grammatical relation. Such markers can be added after the stem (suffixed), before the stem (prefixed), also inserted into the stem (infixed). This is not a special property of grammatical relations, it is a a general property of morphology, that whatever a particular morpheme indicates can be put in a number of different places. Ultimately, what you want to know is, what are the possible ways of adding things to words to make variant forms of words. There are very many.
You can also signal grammatical relations by adding other words, before or after the thing being "marked". Rather than adding suffixes to a noun, Japanese does this with short words after the phrase being marked. Where Finnish has an ablative case suffix, English marks the same grammatical relation with prepositions.
As you noted, word order is very often a marker of grammatical relation, to the point that we talk of languages being VSO, SVO or SOV which tells you how subject and object relations can be computed from word order. It is normal that nominal phrases come in a particular order in a sentence, even in languages that have overt markers like suffixes or postpositions, though there are some languages where word order has little relationship to signalling grammatical relation ("free word-order languages"), which thus rely crucially on affixes to signal grammatical relation.
Control of agreement is another means of signalling grammatical relation. Subjects may govern subject-agreement on verbs, or objects may also govern object-agreement. Khoekhoe (Nama) has the interesting property that the subject governs (double) gender agreement on other nominals when SOV order is not observed, so you can identify the subject as the thing which everything else agrees with and which doesn't have two gender markers.