Mater rogat utrum pueri, quia sitim sentiunt, lac an aquam bibere velint.

Because rogat utrum ... velint introduces an indirect question (a reported question). In Latin, indirect questions regularly take the subjunctive. So velint is “(she asks) whether they want …” reported indirectly, not a direct “they want.”

utrum ... an ... is a standard pattern for an indirect “whether … or …” question offering two alternatives:

• utrum = “whether”
• an = “or (rather)” So utrum ... lac an aquam = “whether (they want) milk or water.”
  You can sometimes see just rogat an ..., but utrum ... an ... is especially natural when two explicit options follow.

Both are direct objects of bibere (“to drink”), so they’re in the accusative.

• aqua, aquae (f.) has accusative singular aquam.
• lac, lactis (n.) is a neuter 3rd-declension noun; its accusative singular is lac (neuter nominative = accusative).

Because velint (“want”) commonly takes a complementary infinitive:
bibere velint = “want to drink.”
So the finite verb is velint, and bibere completes its meaning.

pueri is nominative plural, serving as the subject of sentiunt and velint within the indirect question: “whether the boys … feel thirst … want to drink …”
It isn’t pueros because they are not the object of rogat; they’re the ones doing the wanting/drinking.

Because utrum marks the start of the embedded indirect question, and everything that follows (including pueri ... velint) belongs to what she is asking. The main clause is just Mater rogat.

With quia meaning “because,” Latin usually uses the indicative when the reason is presented as a straightforward fact from the speaker/narrator’s point of view: “because they feel thirst.”
You can find subjunctive with causal conjunctions when the reason is presented as alleged/subjective, but the default with quia is indicative.

Literally it’s “they feel thirst.”
sitim is accusative because sentire takes a direct object: you “feel” something. sitis, sitis (f.) is a 3rd-declension noun; accusative singular is sitim.

Commas in Latin editions are mainly editorial aids for reading; ancient manuscripts didn’t use modern punctuation consistently. The commas here simply mark quia sitim sentiunt as a parenthetical explanatory clause: “the boys, because they’re thirsty, …”

Either order is grammatically possible. The order often reflects emphasis or what’s expected first. Putting lac first can imply it’s the first option considered (or perhaps the more likely/foregrounded option in context). The key structure is X an Y.

Not with the same meaning.

• -ne introduces a neutral yes/no question (“Is it the case that…?”).
• num typically expects the answer “no.”
  But here the question is not yes/no; it’s an alternative question (“milk or water?”), which is exactly what utrum ... an ... expresses.

It’s complete as-is. rogare can take:

• a person asked (accusative): e.g., mater pueros rogat = “the mother asks the boys”
• or a question/thing asked (often via indirect question): mater rogat utrum... = “the mother asks whether...”
  Here the thing being asked is the entire utrum-clause, so no direct object person is necessary.