Learn Hardcore Polish: Trzeba nacisnąć ten przycisk, gdy winda zatrzyma się między piętrami. - One must press this button when the elevator stops between floors.

Trzeba is an impersonal modal expression meaning one must or it is necessary to. It has no explicit subject and is followed by an infinitive (nacisnąć). By contrast, Musisz is a second-person singular form, directly addressing the listener (you must), which changes the tone from general necessity to a direct imperative.

Why is ten przycisk in the same form for nominative and accusative?

Przycisk is a masculine inanimate noun. In Polish, the nominative and accusative forms of masculine inanimate nouns (and their modifiers, like ten) are identical. Here ten przycisk functions as the direct object of nacisnąć, so it takes the accusative, but the form matches the nominative.

Why is the perfective infinitive nacisnąć used instead of the imperfective naciskać?

The perfective aspect (nacisnąć) denotes a single, completed action (pressing the button once). The imperfective (naciskać) implies repetition or an ongoing process (to be pressing). Since the instruction refers to one decisive press, the perfective is required.

Why is gdy used here instead of kiedy, and can they be swapped?

Both gdy and kiedy translate to when. Gdy is often preferred in written or formal contexts for simultaneous actions, while kiedy is more neutral and common in questions. In most cases you can interchange them:
Kiedy winda zatrzyma się między piętrami, trzeba nacisnąć ten przycisk

Could you say kiedy winda się zatrzyma instead of gdy winda zatrzyma się?

Yes. Polish word order is flexible. The reflexive pronoun się can attach after the verb (zatrzyma się) or precede it (się zatrzyma). Placing the time clause first is also fine:
Kiedy winda się zatrzyma między piętrami, trzeba nacisnąć ten przycisk

Why is między piętrami in the instrumental case?

The preposition między (between) requires the instrumental case when indicating a position between two or more points. Piętro (floor) in its plural instrumental form is piętrami, so między piętrami means between floors.

What role does the reflexive pronoun się play in zatrzyma się?

Się makes zatrzymać (to stop something) reflexive, turning it into zatrzymać się (to stop oneself). Here the elevator itself comes to a stop. Reflexive verbs in Polish often indicate that the subject performs the action on itself.

Can the sentence be rephrased to start with the time clause without changing its meaning?

Yes. You can front the time clause for emphasis or style. For example:
Gdy winda zatrzyma się między piętrami, trzeba nacisnąć ten przycisk.
The overall meaning remains the same.

Is it possible to use należy instead of trzeba, and does it change the nuance?

Yes. Należy is another impersonal modal verb meaning one should or it is necessary to, with a slightly more formal tone.
Należy nacisnąć ten przycisk, gdy winda zatrzyma się między piętrami conveys the same instruction in a more official register.