No, the similarity is almost certainly accidental. This kind of coincidental similarity is pretty common, especially in short words like ad.
Latin ad "to, near, at" has cognates in several other branches of Indo-European, including Celtic (Old Irish ad-), Phrygian (αδ-), and Germanic -- English at is among the latter. It appears to go back to a Proto-Indo-...
Your idea is correct.
Lewis-Short is not terribly clear:
added in a direct question, as an interrogation mark, to the first or principal word of the clause
but, if you know German, Georges is much clearer: ‑n(e) is attached to the focus of the question, therefore mostly at the beginning of the sentence.
To answer your question completely, a ...
Both ecce and en can be used with verbs/whole clauses in addition to nouns, to draw attention to some fact rather than an object. So, both ecce [or en] leo edit and ecce [en] leo edens (or ecce [en] leonem edentem) would work.
For ecce used in this way:
ecce, Apollo...imperat ut ego illic oculos exuram (Plautus, Men. 840)
discubitum noctu ut imus, ecce ad ...
Another option (in addition to the several excellent ones in answers so far) is to use (in) loco + gen., as in the phrase in loco parentis "as a parent, in the position of a parent".
Lewis and Short (part IID of the entry) give a number of examples of this usage, such as:
“in uxoris loco habere,” Ter. Heaut. 1, 1, 52: "to consider as a wife"
“in liberum ...
I should like to extend @brianpck's answer by providing two further suggestions.
1. A neat way to express this is by using qua, as in these examples:
— Ad hoc stipatum tribunal, atque etiam ex superiore basilicae parte qua feminae qua viri et audiendi — quod difficile — et — quod facile — visendi studio imminebant. [Pl. Sec. ep.6, 33]
— nam gladiatoribus qua ...
A common and classically attested way of saying "to perform the role of X" is munere X fungi, where "X" is an adjective or genitive noun. Here is an example:
fungar enim iam interpretis munere, ne quis me putet fingere. (Cicero, Tusculanae Disputationes, 3.41)
I will now play the role of an interpreter, lest anyone accuse me of making ...
This answer only concerns Latin; I will leave Greek to others.
Vocative is not the way to go here.
It is used for addressing the god, not for such exclamations.
(At least I have never seen it in such use.)
I would swear using pro (or proh) and a nominative.
See, for example, these search results for the second phrase.
Ne...quidem can most definitely surround nouns in cases other than nominative:
Apuleius, Metamorphoses 9.27 (genitive):
non sum barbarus nec agresti morum squalore praeditus nec ad exemplum naccinae truculentiae sulpuris te letali fumo necabo ac ne iuris quidem seueritate lege de adulteriis ad discrimen uocabo capitis tam uenustum tamque pulchellum ...
A lot of grammars describe the characteristics of sentences involving the particles μέν and δέ. A good example of that is A Greek Grammar for Colleges by Herbert Weir Smyth. In spite of a detailed analysis of the subject, he didn't address this specific question. The same goes for a lot of other grammars, which leads me to believe that it wasn't that common ...
The easy part of your question is the part about Latin. “si” is simply a literal translation of εἰ.
The difficult part is why the Greek original uses εἰ (“if”) when it clearly intends “that”. There are situations where classical Greek uses εἰ is this way. Liddell and Scott write:
after Verbs denoting wonder, delight, indignation, disappointment,
Although it could be read as whether, translations are almost1 consistent in translating these particular instances of si as that.
There are a couple of meanings of si that are equivalent to quod (that) according to L&S. The one I find more applicable here is I.B.2:
In particular, in substantive clauses, to denote a doubtful assumption or future event (...
(This answer may not be satisfying, and I'm happy to have it supplanted in the future! But for now…)
I've never heard of a "γε causal", nor have any of the classicists I've talked to. A Google search brings up physics papers talking about "gamma-epsilon geodesics", which seem to be something in relativity, but no relevant results about Ancient Greek.