Она будет исполнять дуэт, если только боязнь сцены не будет мешать.

Because this is the future tense of an imperfective verb.

• исполнять is imperfective
• imperfective future is formed with будет + infinitive

So:

• будет исполнять = future of исполнять

Russian does this regularly:

• будет читать = will be reading / will read
• будет работать = will be working / will work

If the verb were perfective, Russian would usually use a single-word future instead.

Дуэт is the direct object, so it is in the accusative case. But for many masculine inanimate nouns, the accusative singular looks exactly like the nominative singular.

So:

• nominative: дуэт
• accusative: дуэт

That is why the form does not change.

Because мешать is also an imperfective verb, and it also needs the imperfective future form:

• мешать → будет мешать
• with negation: не будет мешать

So both parts of the sentence refer to the future:

• Она будет исполнять дуэт
• если только боязнь сцены не будет мешать

This is normal in Russian. Unlike English, Russian often uses the future tense in an if-clause when the meaning is future.

Compare:

• English: if stage fright doesn’t interfere
• Russian: если ... не будет мешать

No, it is not a double negative in the English sense.

Here если только introduces a condition with a nuance like:

• provided that
• as long as
• sometimes close to unless

So:

• если только боязнь сцены не будет мешать

means something like:

• provided stage fright doesn’t interfere
• unless stage fright interferes

The не belongs to the verb будет мешать and simply negates it.

Сцены is genitive singular.

That is because боязнь often takes the genitive:

• боязнь чего? = fear of what?
• боязнь сцены = fear of the stage / stage fright

So this is a normal Russian pattern:

• страх темноты = fear of the dark
• боязнь высоты = fear of heights
• боязнь сцены = stage fright

It is singular, not plural.

Very often, yes. The verb мешать normally takes a person in the dative case:

• мешать кому? = hinder whom? / get in the way of whom?

So a fuller version would be:

• Она будет исполнять дуэт, если только боязнь сцены не будет ей мешать.

Here ей = to her

In the original sentence, that dative object is omitted because it is obvious from context that the stage fright would be interfering with her. Russian sometimes leaves this unstated when it is easy to understand.

Yes, it could, but the nuance changes.

• будет исполнять = imperfective future
  This can sound more process-oriented, descriptive, or formal.
• исполнит = perfective future
  This sounds more like a single completed performance or event.

So:

• Она будет исполнять дуэт... = she will be performing a duet...
• Она исполнит дуэт... = she will perform / complete the performance of a duet...

For a single future event, исполнит may sound a bit more event-focused.

Yes. Исполнять is a fairly general and somewhat formal verb meaning to perform.

That makes it a safe choice if you do not want to specify exactly how the duet is performed.

Depending on context, Russian could also use more specific verbs:

• петь if it is sung
• играть if it is instrumental
• танцевать if it is some kind of dance duet

So исполнять дуэт is broad and neutral/formal: to perform a duet.

It is grammatical, but many native speakers might prefer a slightly smoother version, especially by adding ей or by using perfective verbs.

For example:

• Она будет исполнять дуэт, если только ей не будет мешать боязнь сцены.
• Она исполнит дуэт, если ей не помешает боязнь сцены.

The second version is often more compact and more natural for a one-time future event:

• исполнит = perfective
• не помешает = perfective

The original sentence is still understandable and correct.

No. Russian word order is fairly flexible.

The original order is:

• Она будет исполнять дуэт, если только боязнь сцены не будет мешать.

But you could also say:

• Если только боязнь сцены не будет мешать, она будет исполнять дуэт.

Putting the conditional clause first gives more emphasis to the condition.

Also, if you include ей, many speakers may prefer:

• Если только боязнь сцены не будет ей мешать, она будет исполнять дуэт.

or

• Если только ей не будет мешать боязнь сцены, она будет исполнять дуэт.

The meaning stays basically the same, but the focus shifts slightly.

Because если только боязнь сцены не будет мешать is a subordinate clause.

Russian normally separates subordinate clauses from the main clause with a comma.

So:

• main clause: Она будет исполнять дуэт
• subordinate clause: если только боязнь сцены не будет мешать

That is why the comma is required.

Yes, it works, but it can sound a little formal or bookish.

Another very common expression is:

• страх сцены

Both mean stage fright.

A rough difference in feel:

• боязнь сцены = fear of the stage, a bit more formal
• страх сцены = stage fright, often more everyday

So the original phrase is correct, but learners may also hear страх сцены very often.