A mi hermana se le pasan los nervios cuando empieza a hablar.

Yes, they both refer to the same person, and that is completely normal in Spanish.

This is called clitic doubling:

• a mi hermana = the full indirect object
• le = the indirect object pronoun that repeats it

Spanish very often uses both together, especially when the person is specific or already known.

So:

• A mi hermana se le pasan los nervios
  literally has both to my sister and to her

In English we usually would not repeat it, but in Spanish this sounds natural.

Because mi hermana is an indirect object here, not a direct object.

A useful way to see it is:

• los nervios are the thing that pass / go away
• mi hermana is the person they go away from / for

So Spanish uses the indirect object pronoun:

• le = to her

Not:

• la, which would be a direct object pronoun

This sentence is built more like:

• The nerves go away from her than
• The nerves affect her directly

Here it is part of the expression pasársele algo a alguien, which means something like:

• to go away
• to wear off
• to pass
• to stop affecting someone

So:

• se le pasan los nervios = her nerves go away / she gets over her nerves

The parts work like this:

• se = part of the pronominal expression
• le = to her
• pasan = pass / go away

You should learn this as a whole pattern:

• Se me pasa = it goes away for me
• Se te pasa = it goes away for you
• Se le pasa / pasan = it goes away for him/her

Normally, no, not if you want this meaning.

In this sentence, se is part of the idiomatic structure pasársele los nervios a alguien. Without se, the meaning changes or becomes unnatural.

Compare:

• Se le pasan los nervios = her nerves go away
• Pasan los nervios by itself does not naturally mean the same thing

So for this idea, it is best to learn the full pattern:

• A alguien se le pasan los nervios

Because the subject is los nervios, which is plural.

Spanish verbs agree with the subject:

• el nervio pasa
• los nervios pasan

In this sentence:

• los nervios = subject
• pasan = plural verb form

Even though mi hermana appears near the beginning, mi hermana is not the subject.

Yes.

A good way to test it is to ask: what is doing the passing?

Answer:

• los nervios

They are the thing that pass / go away.

Meanwhile:

• a mi hermana
• le

show the person affected by that action.

So the structure is roughly:

• The nerves go away for my sister when she starts speaking.

That is why the verb agrees with los nervios, not with mi hermana.

Spanish often uses the definite article with things that English might leave without one, especially in set expressions and with body-related or emotional states.

So:

• los nervios is the normal way to talk about nerves in this sense

It sounds more natural than just nervios here.

This is similar to other common Spanish patterns where English and Spanish use articles differently.

Because in Spanish, tener nervios or pasársele los nervios is normally expressed with nervios in the plural when talking about nervousness.

So:

• los nervios = nerves / nervousness

Be careful:

• el nervio can mean a nerve
• in some contexts it can also mean cheek / nerve in the sense of boldness

But in this sentence, los nervios is the usual plural expression for being nervous.

Because empezar is followed by a + infinitive when you mean to start doing something.

So:

• empezar a hablar = to start speaking
• empezar a comer = to start eating
• empezar a estudiar = to start studying

After empezar, the a is required in this structure.

Because here cuando is introducing a habitual / general situation:

• whenever she starts speaking, her nerves go away

In Spanish, for repeated or habitual actions like this, you use the indicative:

• cuando empieza a hablar

If you were talking about a future event that has not happened yet, Spanish would normally use the subjunctive:

• cuando empiece a hablar

So the difference is:

• cuando empieza = whenever she starts / when she starts, as a usual pattern
• cuando empiece = when she starts, at some future moment

Because mi hermana is not the subject here. It is the indirect object, so it appears as:

• a mi hermana

The a is required because it marks the person affected by the action.

If you said just Mi hermana at the start, it would suggest she is the subject, but she is not. The subject is los nervios.

Starting with A mi hermana also gives the sentence a natural topic-comment structure:

• As for my sister, her nerves go away when she starts speaking.

Yes, the word order can change, because Spanish is more flexible than English.

For example, these are all possible:

• A mi hermana se le pasan los nervios cuando empieza a hablar.
• Cuando empieza a hablar, a mi hermana se le pasan los nervios.
• Se le pasan los nervios a mi hermana cuando empieza a hablar.

The original version is very natural because it introduces my sister first as the person being talked about.

What usually stays the same is the basic relationship:

• los nervios = subject
• le / a mi hermana = indirect object
• se pasan = the nerves go away