I am not sure I can fully participate in this conversation since Latin is involved and my Latin is not good enough to comment on all the examples you used.
Hoewever here's my two cents:
1. Do sentences exist? Well, yes, as much that is as any abstract creation of man can be said to exist. I prefer the definition about "grammatical" sentences since if we get into what consitutes a "complete" idea we may end up arguing if it takes one sentence or a whole book or even several of them to express an idea completely (if, that is, it is even possible
2. Why is it then that there are cases (I am actually thinking of some examples from Greek) when it's difficult to determine what kind of a sentence we are talking about?
Well, as bellum paxque commented, it is because context governs grammar and because the beauty of language is the multiple layers of meaning it can convey.
This may lead us to question the categories of sentences we have created but not the existence of sentences really. Whether in Example II i.e. the sentence is casual or final (it's final by they way; see bellum paxque's reply and remember that causal sentences refer to the reason that led to the action, not the reason for which the action will happen) the fact remains that it IS a sentence.
If you want to talk about that (the categories) this is a different matter.