Learn Russian: Водитель резко нажал на тормоз, и я едва не облил пассажира кофе. - The driver sharply hit the brake, and I almost spilled coffee on a passenger.

This is the standard Russian construction for an imminent averted action. To say you "almost did" something that you ultimately avoided, you use едва не (or чуть не) plus the perfective past tense verb. Even though there is a не, the meaning is "I barely avoided spilling."

Are едва не and чуть не interchangeable in this sentence?

Yes, they mean exactly the same thing here. Чуть не is extremely common in everyday speech, while едва не can sound a tiny bit more literary or dramatic, but both perfectly fit this sudden situation.

Why do we use the preposition на in нажал на тормоз instead of just using the accusative case directly?

The verb нажать (to press, squeeze, or push) requires на + Accusative when you are applying pressure to a specific surface, button, or pedal. While English says "hit the brake," Russian literally says "pressed on the brake."

What case is кофе functioning as in this sentence?

It is acting as the Instrumental case. The verb облить (to douse, to pour over) takes the Accusative for the person being splashed (пассажира) and the Instrumental for the liquid used. Since кофе is a borrowed, indeclinable noun, its spelling doesn't change, but grammatically it means "with coffee."