"There is no spoon."
If would sound OK if you emphasise «нет» with the intonation. The word order «нет ложки» is usually not used in writing.
Actually, it's quite complex because it depends on what's the new info and what's already known to the listener. We usually put the new info in the end. «Ложки нет» would mean «нет» is the most important word, you're talking about an absence. «Нет ложки» usually means «ложки» is the new info: i.e. listener knows something is missing, and you tell it's spoon that is missing. So «нет ложки» gets to mean 'it's the spoon that's missing'; and it's not the meaning conveyed by this sentence.
However, in speech, the intonation can 'override' the word order. You can emphasise «нет ложки» and it will mark «нет» as the most important word.
So it's hard to say exactly that «this is wrong». No, it's not exactly wrong, it can be correct if you change the intonation or the meaning.
English 'there' in "There is" sentences is a dummy adverb that is required by the grammar, it doesn't really convey any real meaning. If you need to add a location, you add it after the sentence:
- There is no spoon here. ('Here' conveys the location, 'there' is just a dummy adverb that doesn't have a real meaning.)
- There are no spoons in my house. ('In my house' is the real location, 'there' doesn't mean anything.)
'There' exists just because English doesn't allow saying "No spoon is". It doesn't mean anything, it's just there because otherwise the English sentence would be ungrammatical.
Since Russian grammar is different from English, Russian doesn't require any dummy adverbs. Of course, you could add «там» if you wanted, «Там нет ло́жки», but this would be closer to the English sentence "There is no spoon there", and you’d lose the reference to the Matrix if you translate it this way.
You'd notice that Russian also doesn't require other dummy things, like dummy pronouns (they say that... = говорят, что... [when relaying a rumour]; it's getting dark = темнеет [when 'it' doesn't reference a specific object that is getting dark]).