Learn French: J'ai déjà caché le cadeau de peur qu'elle trouve la surprise. - I have already hidden the gift for fear that she might find the surprise.

There can be, but it isn't mandatory! Expressions of fear like de peur que or de crainte que are classic triggers for the ne explétif in formal, written French (which would be de peur qu'elle ne trouve). However, because this ne doesn't actually make the sentence negative, it is almost always dropped in modern, everyday spoken French.

Is trouve in the present indicative or the subjunctive here?

It is in the subjunctive mood. The phrase de peur que (for fear that) always triggers the subjunctive because it deals with a hypothetical, feared outcome rather than a certain fact. For regular -er verbs like trouver, the subjunctive (qu'elle trouve) just happens to be spelled exactly like the present indicative.

Why is déjà placed exactly where it is in the sentence?

In the passé composé, short and commonly used adverbs—like déjà, souvent, toujours, and bien—must be placed directly after the conjugated auxiliary verb (ai) and before the past participle (caché). You cannot put them at the end of the clause like you often do in English.

Could I just say parce que j'ai peur qu'elle trouve instead of using de peur que?