Learn Spanish (Spain): ¿Qué le pasa a tu hermana?

While pasar does mean 'to pass', it is also the most common way to say 'to happen'. Literally, this sentence asks 'What is happening to your sister?', which is the standard Spanish way to say 'What's wrong with your sister?'.

Why is the word le here if we already state it is about tu hermana?

In Spanish, when an action happens to someone, you almost always include the indirect object pronoun (here, le meaning 'to her'). You use it even when you name the exact person later in the same sentence (a tu hermana). It sounds repetitive in English ('What is happening to her, to your sister?'), but it is completely normal and required in Spanish.

What is the purpose of the a right before tu hermana?

The a simply means 'to'. Because the literal meaning of the phrase is 'What is happening to your sister?', you need the a to show who is receiving the action.

Can I translate 'What's wrong with...' word-for-word as ¿Qué está mal con...?

People would understand you, but it sounds like a very unnatural, direct translation from English. ¿Qué le pasa a...? is the natural set phrase you should memorize for 'What's wrong with...' or 'What's the matter with...' someone.