Yes, it's the ergative. As to whether it's an "ending," the short answer is "yes." Long answer is: "Debatable."
ने marks the subject / doer when there is a transitive verb. I dunno if you'd say, therefore, that the /ne/ is part of a "case ending" or just a "particle" that marks. I would say the latter, as this is the custom in North Indian language pedagogy.
There is some variation on whether the /ne/ is written attached to or detached from the subject. I believe the current standard in Devanagari-Hindi is to attach it to the personal pronouns. Rest assured, however, that it is detachable! :) In light of this, it may be best to think of what precedes the /ne/ as the piece that's in a particular "case." Up to you.
As I mentioned in the other comment, I think teachers / describers of Hindi have usually found it most expedient just to say that: 1) /ne/ is needed to mark the subject when there is a transitive verb 2) /ne/, like [other] postposition, puts the preceding noun into the oblique case... 3) ...with the exceptions being the pronouns—whose forms one should check on.
I personally think of /ne/ as a postposition meaning "by". As in: Whom was this done by? By me, by you, by Raj, etc.
Off topic: I don't know if you're a linguist at all. (I am not per se, and I suspect you're a lot more knowledgeable about language grammar generally than me). But you may enjoy this puzzle (for which I have no answer):
In Punjabi, the first and second person, singular and plural, pronouns are not marked with /ne/ in similar situations. Only third-person nouns and pronouns receive /ne/.
I suspect the simply has to do with the fact that /ne/ is not absolutely needed to discern meaning in these cases. In Hindi, it's not needed for comprehension, it merely helps.
I'd love to hear some theories or explanations though. Perhaps /ne/ was historically developed for the third person only, for example, but later taken on in Hindi for the 1st and 2nd person.
Actually, in Punjabi, you can also choose not to use the the particle "ne" with the ergative form of third person pronoun, "Os". Such forms also exist for اسیں، تسیں and plural اوہ. They would be, اساں، تساں، and اونہاں. These forms are not at all mandatory. They depend on dialects and are more often arbitrary. Note the famous, "جس لہور نہیں ویکھیا اوہ جمیا ہی نہیں".
Yes, good examples! But while not using /ne/ in those cases, one has transformed the pronoun. You wouldn't just say "oh" (third person) without adding ne or transforming it to "os" (would you?).
Whereas what I was referring (not being as complete as you in my comment) to was how the first and second-person forms just sit there, with no /ne/, in what looks to be the "direct case" form.
For transitive verbs (verbs which can take direct objects), the verb conjugates with its object(s) in the past tense. If there are no objects in a particular sentence like this one, it is in the masculine singular form.
For intransitive verbs (verbs which cannot take direct objects), the verb always conjugates with the subject.
No. Since खाना is a transitive verb, its form in the past tense (खाया/खाई/खाए) depends on the gender and number of its objects (and not the subject). Eg: क्या तुमने केले खाए?, क्या तुमने केला खाया ?, क्या तुमने रोटी खाई ?
When there is no object like in the sentence 'क्या तुमने खाया?', you use the masculine singular form.
In Hindi, unlike in English, transitivity is an innate property of a verb. So, a transitive verb is one which can have direct objects. It doesn't need to have a direct object in every sentence.
When it has no direct object in a particular sentence like this one, it is used in the singular masculine form (for tenses where transitive verbs take their objects' gender).