Translation:The cat is between the bed and the table.
So the distinction in Latin between "inter" (between) and "intra" (within) developed out of a single root that denoted both. "Inter" is the older of the two, and likely was ambiguous between "between" and "inside" at first. At some point a root "interus, intera" emerged that had similar meaning as "inter", with some ambiguity between "between" and "inside", but leaning towards "inside"; this root gave rise to words like "interior" and "internus", with a meaning of "inside", before subsequently developing into "intra" and losing the meaning of "between"; meanwhile, "inter" kept on truckin' and eventually lost any meaning of "inside", likely due to pressure from the emerging of "interus -> intera -> intra".
I don't think they are either, and your answer should probably be accepted. However (and maybe it's just a regional thing), in this particular instance I would probably use "the bed and the table". For reasons I can't explain, your way feels slightly awkward to me.
Again, though, it's likely just regional differences.
That's actually the regular pattern for feminine singular nouns ending on "-ь": to form the instrumental case, you just append a "-ю" at the end. Болью, плетью, сетью, шалью.
I can think of two exceptions to this rule: "мать" and "дочь". They form their instrumental case from their archaic forms "матерь" and "дочерь" - thus resulting in "матерью" and "дочерью" respectively.
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Prolly just one of those sentences where the number of possible translations is so big that some omissions are simply inevitable. "Стол" can very much be used to refer to a "письменный стол" - so 'desk' should be accepted.
To the unfortunate person who gets it wrong for the same reason and ends up reading this: would you be so kind as to report this one?
Oftentimes, though not necessarily always, Russians put the key information at the end of the sentence. When I read "A cat is between the bed and the table", I get the sense that the original question was, "What's between the bed and the table?", with the key information being the "what" (a cat).
So, for "A cat is between the bad and the table", I would argue "Между кроватью и столом находится кот."
The cat is between a bed and a table. A cat is between a bed and a table.
Would probably just be the same - Кот находится между кроватью и столом.
There isn't really a very big difference here, understanding is driven a lot by context.