“On how serious the mistakes are, or whether you are free to correct them, or your mood, or something?”
No, Sky, I don’t think that construction is an elliptical sentence. I would call it a series of sentence fragments compounded into a question. By definition, a sentence fragment is a word, phrase, or clause that usually has the intonation of a sentence but lacks the grammatical structure that’s usually found in the sentences of formal written composition; in short, unlike a bona fide sentence, a sentence fragment doesn’t have a complete thought. (The sentence fragments in that example of yours are, of course, “on how serious the mistakes are,” “whether you are free to correct them,” and “your mood, or something.”)
A sentence fragment like your example is usually said by the speaker or writer as an afterthought or follow-through to a previous statement, and that speaker or writer assumes that the listener or reader would readily understand its meaning and context despite the incompleteness of the thought expressed.
An elliptical sentence, in contrast, is a sentence construction that gracefully knocks off certain words and phrases (often repetitive ones), taking it for granted that the reader or listener would just mentally fill in the gaps with the missing grammatical elements. The knocked-off words and phrases form “holes” in the sentence that are called ellipses (the singular form of this word is “ellipsis”). As a rule, an elliptical sentence consists of two independent clauses, one containing the grammar elements the other has left out. The independent clause with the missing elements is the elliptical clause—an abbreviated adverb clause stripped of its subject and verb.
The five most common ways of forming elliptical sentences are as follows:
(1) The routine omission of “that” in modifying clauses, particularly in spoken English. This is the most familiar use of the ellipsis. Example: “They knew […] two years would be the shortest time […] they would need to subdue the enemy forces.” (Normal form: “They knew that two years would be the shortest time that they would need to subdue the enemy forces.”)
(2) Elliptical noun phrases. Example: “Jennifer asked for the pink blouse but the salesclerk gave her the red […].” (Normal form: “Jennifer asked for the pink blouse but the salesclerk gave her the red blouse.”)
(3) Ellipsis of the verb and its objects or complements. Example: “The beleaguered Supreme Court chief justice would fight it to the very end if he could […].” (Normal form: “The beleaguered Supreme Court chief justice would fight it to the very end if he could fight it to the very end.”)
(4) Medial (middle) ellipsis. Example: “Arlene will take care of the girls and Eduardo […], the boys.” (Normal form: “Arlene will take care of the girls and Eduardo will take care of the boys.”)
(5) Ellipsis of clause. Examples: “They can leave now if they want […].” (Normal form: “They can leave now if they want to leave now.”) Certain elliptical clauses, however, need a comma to indicate that some words have been intentionally left out; otherwise, confusion might arise. Properly elliptical: “My tour group chose Paris; theirs, Rome.” Improperly elliptical: “My tour group chose Paris; theirs Rome.” (Normal form: “My tour group chose Paris; their group chose Rome.”)
By now I think you can already see the big difference between a sentence fragment and an elliptical sentence: a sentence fragment is, as the term implies, a fractured statement with an incomplete thought, while an elliptical sentence is a streamlined, more concise sentence that manages to deliver a complete thought despite dropping some of its words and phrases.
