Learn German: Habt ihr euch die Hände gewaschen?

euch is a dative reflexive pronoun here. In German, when you perform an action on a part of your own body, you use a reflexive pronoun to show who is affected by the action. Because the sentence already has a direct object (the thing being washed, which is die Hände), the reflexive pronoun must be in the dative case.

Why does it say die Hände (the hands) instead of eure Hände (your hands)?

In English, you say you wash "your hands". In German, you don't need a possessive word like eure because the reflexive pronoun euch already makes it perfectly clear whose hands are getting washed. Instead, you just use the definite article die (the).

Is euch accusative or dative in this sentence? They look exactly the same!

It is dative! You are right that for ihr (you all), the accusative and dative pronouns are both euch. We know it is dative here because the sentence already has a direct object in the accusative case: die Hände (the hands).

Why is the helping verb Habt used instead of Seid for the past tense?

The verb waschen (to wash) uses haben to form the past tense because it takes a direct object (you are washing something). You only use sein (to be) as a helping verb for verbs that show movement from A to B (like going or driving) or a change of state (like waking up).