Espero que te vaya bien en la entrevista.

Because Espero que... normally triggers the subjunctive in Spanish.

• va = present indicative
• vaya = present subjunctive

After verbs of wishing, hoping, wanting, or emotion, Spanish often uses the subjunctive in the next clause:

• Espero que vengas.
• Quiero que me llames.
• Me alegra que estés aquí.

So in Espero que te vaya bien, the speaker is expressing a hope about what will happen, not stating a fact.

It literally comes from the expression irle bien a alguien, which means to go well for someone or to turn out well for someone.

So:

• te = to you / for you
• vaya bien = goes well

Together, que te vaya bien means something like:

• that it goes well for you
• that things go well for you
• that you do well

In this sentence, it means I hope the interview goes well for you.

Te is the indirect object pronoun for you in the informal singular form (tú).

The expression is:

• irle bien a alguien = to go well for someone

So:

• me va bien = it goes well for me
• te va bien = it goes well for you
• le va bien = it goes well for him/her/you (formal)

In your sentence, te shows who the good outcome is for: you.

It is informal singular, because it uses te, which corresponds to tú.

If you were speaking formally to one person in Spain, you would usually say:

• Espero que le vaya bien en la entrevista.

If you were speaking to several people in Spain, you would say:

• Espero que os vaya bien en la entrevista.

So te tells you the speaker is talking to one person informally.

Yes. Vaya is the present subjunctive form of ir.

Here are the present subjunctive forms of ir:

• yo vaya
• tú vayas
• él/ella/usted vaya
• nosotros/nosotras vayamos
• vosotros/vosotras vayáis
• ellos/ellas/ustedes vayan

In your sentence, the subject is really la entrevista or more generally everything related to it, so Spanish uses the third person singular form: vaya.

Because en la entrevista here means in the interview / during the interview / in that situation.

It points to the context where you hope things go well.

• Espero que te vaya bien en la entrevista. = I hope things go well for you in the interview.

If you say para la entrevista, that usually sounds more like for the interview, meaning preparation or purpose:

• Necesitas un traje para la entrevista. = You need a suit for the interview.

So:

• en la entrevista = during the interview itself
• para la entrevista = for the interview, in preparation for it

Yes, that is understandable, but Espero que te vaya bien en la entrevista sounds more natural.

Why?

Because irle bien a alguien en algo is a very common pattern:

• Te irá bien en el examen.
• Le fue bien en el trabajo.
• Nos va bien en clase.

If you say la entrevista te vaya bien, you are making la entrevista more directly the thing that is going well, which is not impossible, but it is less idiomatic than the usual te vaya bien en la entrevista.

Yes, and it changes the nuance slightly.

• Espero que te vaya bien en la entrevista = I hope things go well for you in the interview / I hope you do well
• Espero que la entrevista salga bien = I hope the interview turns out well

So:

• ir bien focuses more on how it goes for you
• salir bien focuses more on the result or outcome

Both are common, but your original sentence is especially natural when speaking directly to the person who is going to the interview.

Because Spanish usually does not need subject pronouns when the verb ending already shows who the subject is.

• Espero already means I hope
• So yo espero is possible, but usually unnecessary

Spanish often omits pronouns unless the speaker wants emphasis, contrast, or clarity.

Compare:

• Espero que te vaya bien. = neutral, normal
• Yo espero que te vaya bien. = more emphasis on I

Usually, without more context, most people will understand la entrevista as a job interview. That is the most common interpretation.

But entrevista can also mean other kinds of interviews, such as:

• a media interview
• a school interview
• an immigration interview
• any formal question-and-answer meeting

So the word itself is broad, but in everyday use this sentence very often refers to a job interview.

Yes. Que te vaya bien is a very common expression on its own.

It can mean things like:

• Hope it goes well for you
• Take care / all the best
• I hope everything goes well

People often use it as a friendly closing or good wish, similar to:

• Que te vaya bien.
• Que os vaya bien.
• Que le vaya bien.

In your sentence, Espero makes the wish explicit, but the shorter expression is also very natural in everyday Spanish.