Learn Russian: Благодаря новому плану враг остался равнодушным. - Thanks to the new plan, the enemy remained indifferent.

The verb остаться (to remain) is a verb of state. In Russian, when verbs like оставаться/остаться, оказываться/оказаться, or быть connect a subject to its state or condition, that state takes the Instrumental case. Here, равнодушным describes the condition the enemy remained in.

What case does благодаря require, and could I use из-за here instead?

The preposition благодаря (thanks to) always requires the Dative case, which is why we see новому плану. You wouldn't use из-за (because of) here, because благодаря implies a positive or successful outcome, whereas из-за usually introduces a negative cause or obstacle.

How does the literal meaning of равнодушный translate to 'indifferent'?

It is a compound word formed from равный (equal) and душа (soul or spirit). In Russian, having an 'equal soul' means you are emotionally detached, apathetic, or indifferent to what is happening.

Why use the perfective verb остался instead of the imperfective оставался?

The perfective остался focuses on the result achieved by the new plan. It presents the enemy's indifference as a completed outcome. Using the imperfective оставался would focus on the ongoing background duration of their indifference rather than the factual result.