I know generally about the concept behind the partitive but one aspect doesn't seem to be explained anywhere online. The partitive occupies the same structure that denotes nominative and accusative cases, among others, which doesn't make much sense to me. Couldn't a partitive noun be the subject or the object, and since it's marked with partitive it ISN'T marked with either of those so how can you tell which it is in the sentence? Some languages exhibit the partitive with prepositional phrases, which is essentially the same I guess, and Finnish also uses up a case ending on it.

Is it an unwritten assumed rule that anything in the partitive must be the object of the sentence?

That would line up with a lot of Russian mentality since only mass, inanimate nouns really ever take the partitive, and those will theoretically never be subjects.

It's just occurred to me, that Russian partitive can be accompanied by a "determining" noun, which will take on the case endings for the whole phrase.
(a) чаш-ка,-ку,-кой,-ки чаю

A few questions remain though:
-What is the difference between кашка чаю and чашка чая?

-What about the times where there isn't a determining noun to mark the phrase? How can one tell how the чай fits grammatically? Must we just assume BY CONTEXT that выпить will probably have a liquid based object?
(b) Выпить чаю/чая/чай

-What differences are there between these three?

Is the partitive an noun-attributive or a verb-agreeing case? Meaning, for instance, Genitive is basically used to affect ~a noun~ with a noun, it's unrelated to the verb layout. Whereas Accusative is directly interworking with the sentence's verb. (a) and (b) show it being both types.

-Is modern partitive simply represented by genitive endings, even though the same trick is at work? ( Выпить чая )