Vaya donde vaya, mi madre siempre busca una locutora que hable despacio.

This is a very common Spanish pattern: verb + relative word + same verb. It expresses no matter ... / whatever / wherever.

So:

• Vaya donde vaya = wherever she goes
• Haga lo que haga = whatever she does
• Sea como sea = however it is / be that as it may

Both vaya forms here come from ir and the repetition is part of the structure, not a mistake.

Because the sentence is not talking about one specific place she goes to. It means any place at all, or wherever.

Spanish often uses the subjunctive with:

• indefinite or unknown places, people, or things
• expressions like wherever, whoever, whatever
• concessive ideas like no matter where

So donde va would sound like a more specific or factual statement, while donde vaya fits the idea of wherever she may go.

Because here donde is a relative adverb, not a question word.

• dónde = where? in direct or indirect questions
  
  • ¿Dónde vive?
  • No sé dónde vive.
• donde = where/wherever in relative clauses
  
  • Vive donde quiere.
  • Vaya donde vaya...

So in this sentence, no accent is needed.

Yes. With verbs of movement like ir, Spanish can use:

• donde
• adonde
• a donde

All can refer to destination.

In modern everyday Spanish, especially in expressions like this, donde vaya is very common and completely natural. Adonde vaya may sound a bit more explicitly directional, but the meaning is basically the same here.

Because mi madre is looking for any announcer with that quality, not one specific, identified announcer.

Spanish often uses the subjunctive after que when the noun being described is:

• wanted
• searched for
• unknown
• hypothetical
• not yet identified

So:

• busca una locutora que hable despacio = she is looking for a female announcer who speaks slowly, but not a specific one

Compare:

• Busca una locutora que hable despacio.
  She wants to find one like that.
• Busca a la locutora que habla despacio.
  She is looking for the specific announcer who speaks slowly.

That contrast between subjunctive and indicative is very important in Spanish.

Because the speaker means any female announcer of that type, not a particular known one.

• una locutora = a nonspecific announcer
• la locutora = a specific announcer already known to the speaker/listener

This matches the subjunctive hable, since both point to someone not yet identified.

Locutora is the feminine form of locutor.

In Spain, it often refers to a woman who works as:

• a radio announcer
• a presenter
• a broadcast voice
• a voice-over speaker

Depending on context, English translations can vary a bit. It does not always mean exactly the same thing as speaker in the general sense.

Both can mean slowly, but despacio is usually more natural in everyday speech, especially with hablar.

• hablar despacio = the most common, idiomatic choice
• hablar lentamente = correct, but a bit more formal or deliberate

So que hable despacio sounds very natural.

Not exactly. Even though it uses the present subjunctive, the meaning is more like a general truth or habitual situation:

• wherever she goes
• no matter where she goes

It is not limited to what she is doing right now. It describes a repeated pattern.

If you wanted a past version, Spanish could use something like:

• Fuera donde fuera...
• Adonde fuera...

depending on style and context.

The comma separates the introductory subordinate clause from the main clause:

• Vaya donde vaya, = introductory idea: wherever she goes
• mi madre siempre busca una locutora que hable despacio = main statement

In English, we often do the same:

• Wherever she goes, my mother always looks for...

So the comma is very natural here.

Yes. A few alternatives are possible, though some are more formal or literary than others:

• Adonde vaya, mi madre siempre busca una locutora que hable despacio.
• Vaya adonde vaya, mi madre siempre busca una locutora que hable despacio.
• Dondequiera que vaya, mi madre siempre busca una locutora que hable despacio.

The original Vaya donde vaya is very idiomatic and common. It is probably the most useful pattern to learn first.