Learn Hardcore German: Die Mitarbeiterin sagt, im Nachbarhaus gebe es keinen Kopierer. - The female staff member says there is no copier in the neighboring house.

Because this is reported speech using the subjunctive I (Konjunktiv I). In indirect/reported statements, German often uses Konjunktiv I to signal that the speaker is relaying someone else’s words without endorsing them. The indicative would be gibt, but here we have gebe to mark reported speech: Die Mitarbeiterin sagt, … es gebe …

When would I use gäbe instead of gebe?

Gäbe is Konjunktiv II. You use it for hypotheticals or when Konjunktiv I looks identical to the indicative and you want to avoid ambiguity. In reported speech, gebe is the default here because it’s clearly distinct from gibt. Gäbe can also imply doubt or unreliability: Er sagte, es gäbe dort keinen Kopierer suggests distance or skepticism. For a pure hypothetical: Wenn es im Nachbarhaus einen Kopierer gäbe, …

Is the clause after the comma a subordinate clause? There’s no dass.

It’s indirect speech without dass, which keeps main-clause word order (verb in second position). With dass, you’d get verb-final: Die Mitarbeiterin sagt, dass es im Nachbarhaus keinen Kopierer gibt. Without dass, it’s: …, im Nachbarhaus gebe es … (finite verb in 2nd position after the fronted phrase).

Why is the order gebe es and not es gebe?

Because im Nachbarhaus has been fronted into first position. German main clauses are verb-second: first field (here, im Nachbarhaus) + finite verb (here, gebe) + the rest (here, es …). If you don’t front the place phrase, you can say Es gebe im Nachbarhaus keinen Kopierer—also correct, just a different emphasis.

Could I just say Die Mitarbeiterin sagt, im Nachbarhaus gibt es keinen Kopierer?

Yes, that’s common in everyday speech (indicative in indirect speech). However, in careful or journalistic German, Konjunktiv I (gebe) is preferred to mark reportedness and neutrality.

Why keinen Kopierer and not kein Kopierer?

Because es gibt governs the accusative for the thing that exists/doesn’t exist. Kopierer is masculine; the accusative of kein for masculine singular is keinen. Think: nominative ein/kein Kopierer, accusative einen/keinen Kopierer.

What case is im Nachbarhaus, and why?

Dative. Im = in dem. With in, use dative for location (where?) and accusative for motion (where to?). Here it’s a static location, so dative: im Nachbarhaus. For motion, you’d see ins Nachbarhaus (in das).

Do I need the es in es gebe? What is it doing?

Yes. Es gibt is a fixed existential construction meaning “there is/are.” The es is an expletive (dummy subject). In your sentence, even with word-order changes, you keep it: gebe es, es gebe, etc.

What’s the nuance of Nachbarhaus vs alternatives like Nebenhaus or nebenan?

Nachbarhaus: the house next door/the neighboring building (neutral, common).
Nebenhaus: the adjacent building (often within the same complex; can sound a bit more technical/site-specific).
nebenan: adverb meaning “next door.” Example: Im Haus nebenan gibt es keinen Kopierer.

What does Mitarbeiterin tell me?

The suffix -in marks the feminine form: Mitarbeiterin = female employee. Masculine is der Mitarbeiter; plural mixed or all-male is die Mitarbeiter, feminine plural is die Mitarbeiterinnen. The article die here is definite (a specific woman). You could also say Eine Mitarbeiterin sagt, … for “An employee says, …”

How would I write the direct speech version?

Die Mitarbeiterin sagt: Im Nachbarhaus gibt es keinen Kopierer.

Is the comma after sagt required?

Yes. You separate the reporting clause from the reported clause with a comma. If you switch to direct speech, you typically use a colon: Die Mitarbeiterin sagt: …