Learn Spanish (Latin America): Reprobé el examen porque olvidé la direccional. - I failed the exam because I forgot the turn signal.

In much of Latin America (especially Mexico and neighboring countries), la direccional is the standard term for a car's turn signal or blinker. It is used as a feminine noun. Note that if you are exposed to Spanish from Spain, you will usually hear el intermitente instead.

I thought "to fail" was fracasar. Why use reprobé here?

Fracasar means "to fail" in a broad, life or business sense (like a project or a marriage failing). When talking specifically about failing a test, a class, or a grading evaluation, reprobar is the standard verb in Latin American Spanish.

Could I say se me olvidó la direccional instead of olvidé la direccional?

Yes, and it would actually sound a bit more natural in spoken Spanish! Using the accidental "se" structure (se me olvidó) is extremely common in everyday speech because it emphasizes that the forgetting was unintentional (literally, "the turn signal forgot itself to me"). Olvidé is perfectly correct but sounds slightly more direct and takes personal responsibility for the slip-up.

Based on this lesson, how would I express the regret: "If only I hadn't forgotten the turn signal!"?

Because you are expressing an unfulfillable wish about something that already happened in the past, you need ojalá plus the past perfect subjunctive. You would say: ¡Ojalá no hubiera olvidado la direccional!