Learn Russian: Ты потерял свои наушники?

In Russian, when the object belongs to the subject of the sentence (here, Ты - "you"), you use the reflexive possessive pronoun свой (meaning "one's own"). While твои would be understood, свои is the natural and grammatically correct choice here to say "Did you lose your own headphones?".

What case is свои наушники in here?

It is in the Accusative case, because the headphones are the direct object (the thing being lost). Since наушники (headphones) is an inanimate plural noun, its Accusative form looks exactly the same as its Nominative (dictionary) form.

Does the verb потерял tell us the gender of the person being asked?

Yes! Потерял has the masculine past tense ending -л. If you were asking a female if she lost her headphones, you would use the feminine past tense ending -ла: Ты потеряла свои наушники?

Is наушники (headphones) always plural?

Yes, it is typically plural, just like "headphones" in English, because it refers to a pair. You can spot the root word уши (ears) right in the middle of it, combined with the prefix на- (on).