Learn Russian: Капитан не использовал компас в течение этого плавания. - The captain did not use a compass during this voyage.

В течение (meaning "during" or "over the course of") is a fixed compound preposition that always ends in -е. If you write в течении (ending in -и), it becomes a regular noun in the prepositional case, meaning "in the current" (for example, the current of a river).

What case does в течение require for the words that follow it?

It always requires the Genitive case. Here, the neuter noun плавание (voyage) and its demonstrative pronoun это correctly change to этого плавания.

Использовать is a bi-aspectual verb (it can be imperfective or perfective). Which one is it here?

It acts as an imperfective verb here. The preposition в течение highlights the continuous, ongoing duration of the voyage. When describing a state or action (or lack thereof) that spans steadily across a set duration, Russian requires the imperfective aspect.

Why is the direct object компас in the Accusative case instead of the Genitive, since the verb is negated?

While negated verbs often take the Genitive case for their direct objects (the "Genitive of negation"), for specific, concrete, and easily tangible physical objects like a compass, modern Russian usually prefers the Accusative case. Using the Accusative (компас) sounds much more natural here than the Genitive (компаса).