Learn Russian: В верхних ложах не сидит ни один зритель. - Not a single spectator is sitting in the upper boxes.

Russian requires double negation here. To say 'not a single spectator does [action]', you must negate the verb with не (не сидит) and use the emphatic particle ни with the pronoun (ни один).

Why is ни один зритель in the nominative case? Shouldn't negated subjects be in the genitive?

You would use the genitive in a negative existential sentence (for example, using нет to say 'there is not a single spectator'). However, in this sentence, ни один зритель is the active grammatical subject of the verb сидит. When 'not one X' is the subject of a standard verb, it stays in the nominative case.

What is the base form of ложах and is it related to the word for a bed?

The base form is ложа, a feminine noun meaning a theater box or lodge. It is in the prepositional plural here because of the preposition в. Be careful not to confuse it with the neuter noun ложе (a bed or couch), which has different declension endings.

Why is the subject at the very end of the sentence?

While you could easily say Ни один зритель не сидит в верхних ложах, Russian word order uses position for emphasis. Setting the scene first with В верхних ложах allows the sentence to end with the most important, emphatic new information: that absolutely no one is sitting there.