Learn Dutch: We beloven een snelle afhandeling van de terugbetaling. - We promise a fast processing of the refund.

Because afhandeling is a de-word. In Dutch, nouns ending in -ing are always de-words. When an adjective comes before a de-word, it always gets an -e ending, even if it is preceded by the indefinite article een.

What verb does afhandeling come from, and what exactly does it mean?

It comes from the separable verb afhandelen, meaning to handle, settle, or process (a matter or a task). Adding the suffix -ing turns it into the noun afhandeling (the handling or processing).

Is terugbetaling the only word for "refund"?

It is the most common and versatile one, literally translating to "back-payment" (from the verb terugbetalen). You might also see restitutie in highly formal or legal terms and conditions, but terugbetaling is the standard term for general business and customer service.

This sentence uses a heavy noun phrase (een snelle afhandeling van de terugbetaling). How could you rephrase this to use a verb structure instead?

You could say We beloven (om) de terugbetaling snel af te handelen (We promise to process the refund quickly). While the noun phrase in the original sentence is very common in formal customer service emails, everyday spoken Dutch often prefers using verbs with a te construction.