Espero que el micrófono funcione bien en el auditorio mañana.

Because Espero que... expresses a hope/wish, which triggers the present subjunctive in Spanish. The verb funcionar becomes funcione in the yo/él/ella/usted form of the present subjunctive:

• (Indicative) funciona = it works (stating a fact)
• (Subjunctive) funcione = (I hope) it works (not presented as a fact)

After Espero que, Spanish normally uses the present subjunctive even when the action is in the future. Time words like mañana already show the future timeframe:

• Espero que funcione mañana = I hope it works tomorrow
  Using funcionará would change the structure (no longer dependent on Espero que) and would sound like a prediction/statement, not a hope.

In this structure, Espero que + subjunctive means I hope that... or I’m hoping that....
On its own, Espero... can also mean I wait... in other contexts (like Espero el autobús = I’m waiting for the bus), but Espero que is clearly the “hope” construction.

You need que because Spanish uses Espero que + clause to connect what you hope with a full sentence.
Espero el micrófono funcione... is not grammatical. Correct options include:

• Espero que el micrófono funcione...
• Espero que funcione el micrófono... (less neutral word order)

Funcione is present subjunctive (third person singular / usted-form). It comes from:

• infinitive: funcionar
• subjunctive stem: funcion-
• ending: -e (for -ar verbs)

Spanish normally uses a definite article (el/la/los/las) with specific, known things more often than English does. Here it’s a specific microphone (the one to be used tomorrow), so el micrófono is natural.

Here bien means well (describing how it functions). Bueno is an adjective meaning good, used to describe a noun:

• funcione bien = works well (adverb)
• un micrófono bueno = a good microphone (adjective)

It can be understood as “work well in the auditorium,” i.e., in that setting/location. In context, it implies you’re concerned about how it performs there (sound, equipment, setup). If you wanted to emphasize “there (not elsewhere),” you might add emphasis:

• ...funcione bien en el auditorio (y no en otro lugar)

Mostly yes. Auditorio typically refers to an auditorium: a large hall for talks, concerts, presentations, etc. In some places it can also refer to a lecture hall or venue hall depending on context.

Here it means tomorrow, because it’s used as a standalone time word at the end of the sentence.
mañana can mean morning when used with an article or a clearer time expression, for example:

• por la mañana = in the morning
• esta mañana = this morning

Yes. Spanish is flexible with time expressions. These are all natural, with slightly different emphasis:

• Espero que el micrófono funcione bien en el auditorio mañana. (neutral)
• Espero que mañana el micrófono funcione bien en el auditorio. (emphasizes tomorrow)
• Espero que el micrófono funcione bien mañana en el auditorio. (emphasizes tomorrow a bit more)

The accent mark shows the stressed syllable: mi-CRÓ-fo-no.
Without the accent, Spanish stress rules would place stress differently, so the accent is required for correct pronunciation and spelling.

You can, but it changes the meaning slightly:

• funcione bien = that it functions/works well (performance)
• esté bien = that it’s okay / in good condition (state/condition)
  For equipment, funcione bien is usually the best match when you mean “works properly.”