Learn French: Tu es surpris qu'elle ait demandé la mutation ? - Are you surprised she asked for the transfer?

The phrase Tu es surpris expresses an emotion, which requires the verb in the following que clause to be in the subjunctive mood. Because her asking for the transfer happened in the past, before the feeling of surprise, we must use the past subjunctive (le subjonctif passé) rather than the present subjunctive.

How do I know whether to use the auxiliary avoir or être to form this past subjunctive?

The subjonctif passé uses the exact same auxiliary rules as the passé composé. Since the verb demander takes avoir in the passé composé (elle a demandé), you simply conjugate avoir in the present subjunctive (qu'elle ait) and add the past participle (demandé).

Does la mutation refer to a genetic mutation here?

No, while it can mean a biological mutation, in a professional or administrative context une mutation means a job transfer or relocation. It is a very common term in the French working world.

Could I avoid the subjunctive here by using de + infinitive?