"Where do you like to sit?"
Translation:Kie vi ŝatas sidi?
Kie is not the object of the sentence, it's an adverb of place. Also, sidi is an intransitive word, which means it can't have an object. (you can't say "I sit something" in English either)
Also, practically speaking "kien" means "where to" (a destination). For example:
Kie vi kuras? (Where are you running? - e.g. on the road)
Kien vi kuras? (Where to are you running? - e.g. to the park)
This is also possible with other phrases. As Wikipedia explains:
A frequent use of the accusative is in place of al (to) to indicate the direction or goal of motion (allative construction). It is especially common when there would otherwise be a double preposition:
la kato ĉasis la muson en la domo (the cat chased the mouse in [inside of] the house)
la kato ĉasis la muson en la domon (the cat chased the mouse into the house).
"Kie vi ŝatas sidi" = Where do you like to be sitting, static location.
"Kien vi ŝatas sidi" = Where do you like to sit TO. Normally this sentence wouldn't make sense since you don't move when you sit, since it implies you're in transit to another location while performing the sitting action. However, it can actually be used, maybe in a bit of strange context, with the point of sitting in a vehicle that is moving. (Train, plane, etc.)
The question starts with the word "Kie", rendering the inclusion of the word "Cxu" to be unnecessary. Since both words are "question starters" in Esperanto, "Kie" also renders the inclusion of the word "Cxu" to be redundant. The meaning of the word "Cxu" in Esperanto is actually closer to the meaning of the word "Whether" in English. If you wanted to ask about where someone would like to sit using a "Cxu" format, one could ask, "Cxu vi sxatus sidi tie?" to ask whether or not you would like to sit there (which would likely be specified by way of some kind of gesture) or else "Cxu vi sxatus sidi apud mi?" to ask whether or not you would like to sit next to them. An example that more directly uses the word "cxu" the way an English speaker would use whether is "Sxi konsideris cxu iri aux ne iri," which means, "She considered whether to go or not to go."
I appreciate your comprehensive reply. I did realize that "ĉu" was redundant with the presence of "kie"; the issue was more that I did not know how to think about what function "ĉu" performed, because almost all of the Duolingo practice sentences with "ĉu" placed it in the front, as if it was a correlative of the kie, kio, kia type! But now that I can map it to "whether" in my head, I think I even know how to use "ĉu" in the middle of the sentence now.