Si el pájaro vuelve a hacer ruido, moveré la jaula al pasillo.

Because they are two different words:

• si = if
• sí = yes or himself/herself/itself in some contexts

So in this sentence, Si el pájaro vuelve a hacer ruido... means If the bird makes noise again...

The accent mark changes the meaning.

After si meaning if, Spanish normally uses the present indicative for a real, possible future condition.

So Spanish says:

• Si el pájaro vuelve a hacer ruido, moveré la jaula...

Literally, that is structured like:

• If the bird makes noise again, I will move the cage...

This is actually quite similar to English, where we also say:

• If it rains tomorrow, I’ll stay home

  not

• If it will rain tomorrow...

So vuelve is present tense in form, but it refers to a future possibility because it is inside a condition introduced by si.

Because after si for a normal, likely condition, Spanish uses the indicative, not the subjunctive.

So the correct form is:

• Si el pájaro vuelve...

not

• Si el pájaro vuelva...

This is one of the most important Spanish patterns to learn:

• Si + present indicative, then a future/result clause
  
  • Si viene, hablaré con él.
  • Si llueve, no saldremos.

The subjunctive is used in many other kinds of clauses, but not here.

Volver a + infinitive means to do something again.

So:

• volver a hacer ruido = to make noise again
• volver a llamar = to call again
• volver a intentarlo = to try again

This is a very common Spanish structure.

In this sentence:

• el pájaro vuelve a hacer ruido = the bird makes noise again

Because hacer ruido is the normal Spanish expression for to make noise.

• hacer = to do / to make
• ruido = noise

Together:

• hacer ruido = to make noise

English also uses a noun here, so the structure is quite close.

Examples:

• Los niños hacen mucho ruido. = The children make a lot of noise.
• No hagas ruido. = Don’t make noise.

Moveré is the future simple of mover (to move).

For regular future forms in Spanish, you usually add the endings directly to the infinitive:

• mover + é = moveré
• mover + ás = moverás
• mover + á = moverá
• mover + emos = moveremos
• mover + éis = moveréis
• mover + án = moverán

So moveré means I will move.

Yes. That would also be natural.

• Si el pájaro vuelve a hacer ruido, voy a mover la jaula al pasillo.

Both mean roughly the same thing:

• moveré = I will move
• voy a mover = I’m going to move

In everyday speech, ir a + infinitive is very common.
The future simple (moveré) can sound a bit more neutral, firm, or written, but both are normal.

Because a + el contracts to al.

So:

• a + el pasillo → al pasillo

This contraction is mandatory in standard Spanish.

Compare:

• Voy al supermercado.
• Sube al coche.

But this only happens with el, not with la:

• a la cocina — not contracted

Spanish uses definite articles more often than English does.

Here, the articles sound natural because the sentence refers to specific things in the situation:

• el pájaro = the bird
• la jaula = the cage
• el pasillo = the hallway/corridor

Even when English might sometimes drop an article in a similar context, Spanish often keeps it.

They show which syllable is stressed.

• pájaro is stressed on the first syllable: PÁ-ja-ro
• moveré is stressed on the last syllable: mo-ve-RÉ

Without the accent marks, the stress would follow the normal spelling rules and would fall on a different syllable.

So the accents are there to show the correct pronunciation.

It is normal and recommended here because the si clause comes first:

• Si el pájaro vuelve a hacer ruido, moveré la jaula al pasillo.

When the condition comes first, Spanish usually separates it with a comma.

If you reverse the order, you normally do not use a comma:

• Moveré la jaula al pasillo si el pájaro vuelve a hacer ruido.

It can mean both, depending on context.

In Spain, pasillo commonly refers to:

• a hallway
• a corridor
• sometimes an aisle in a shop, theatre, plane, etc.

In this sentence, al pasillo would most naturally mean into the hallway/corridor.

Yes, often you can.

For example:

• Si el pájaro hace ruido otra vez, moveré la jaula al pasillo.

That is also natural.

The difference is mainly stylistic:

• volver a + infinitive = a very common way to say do again
• otra vez = again, added separately

Both are correct, but vuelve a hacer ruido sounds very idiomatic and smooth.