Nadie debe amenazar al vecino.

Because nadie is grammatically singular in Spanish. It means nobody / no one, but it takes a singular verb:

• Nadie debe... = Nobody should/must...
• Nadie sabe... = Nobody knows...

So even though it refers to zero people, Spanish treats it as a singular subject.

Because nadie already carries the negative meaning by itself when it comes before the verb.

• Nadie debe amenazar al vecino. = correct
• No nadie debe... = not correct

A useful rule is:

• If the negative word comes before the verb, you usually do not add no.
• If it comes after the verb, you usually do add no.

For example:

• Nadie vino. = Nobody came.
• No vino nadie. = Nobody came.

Both are correct, but the structure changes.

It can mean either, depending on context. The verb deber often expresses obligation, so it can be translated as:

• should
• must
• ought to

In a sentence like Nadie debe amenazar al vecino, English could render it as either:

• Nobody should threaten the neighbor
• Nobody must threaten the neighbor

In natural English, should is often the smoother choice unless the context is very strict or legal, where must not might fit better.

Because after deber, Spanish normally uses an infinitive.

This is similar to English modal-style structures:

• debe amenazar = should threaten
• puede amenazar = can threaten
• quiere amenazar = wants to threaten

So the pattern is:

• subject + deber + infinitive

Here:

• Nadie = subject
• debe = conjugated form of deber
• amenazar = infinitive

Because a + el contracts to al in Spanish.

So:

• a + el vecino → al vecino

This contraction is mandatory.

Compare:

• Veo al profesor.
• Ayudo al niño.
• Amenaza al vecino.

But there is no contraction with la:

• a la vecina = correct
• not al vecina

That is the personal a. In Spanish, when the direct object is a specific person or is treated as human, you usually add a.

Here, el vecino is a person, so Spanish says:

• amenazar al vecino

not just:

• amenazar el vecino

This is very common with verbs that act on people:

• Veo a María.
• Ayudo al doctor.
• Conozco a tu hermano.

So in this sentence, al vecino is the direct object, and it gets the personal a because it refers to a person.

It can be either, depending on context.

In Spanish, the definite article often appears where English might use:

• the
• a
• or a more generic expression

So al vecino could mean:

• the neighbor in a specific situation
• or a more general idea like one’s neighbor / the neighbor as a type of person

Spanish often uses the singular definite article to talk about people or things in a general way.

If you wanted a clearly plural meaning, you would say:

• Nadie debe amenazar a los vecinos. = Nobody should threaten the neighbors.

Not with the same meaning.

There is an important difference:

• deber + infinitive = obligation
  
  • Debe amenazar = He/She should/must threaten
• deber de + infinitive = probability or assumption
  
  • Debe de amenazar = He/She is probably threatening or must be threatening

So in your sentence, the correct form for obligation is:

• Nadie debe amenazar al vecino.

Yes, but the given order is the most neutral and standard one.

Normal order:

• Nadie debe amenazar al vecino.

You can move elements for emphasis, but then the sentence may sound more marked or literary. For example, fronting the object is possible in some contexts, but the basic learner-friendly pattern is best kept as:

• subject + deber + infinitive + object

Also, debe amenazar works as a close unit, so you generally should not split it in unnatural ways.

A simple guide for Spain pronunciation would be:

• Nadie → NA-dye
• debe → DEH-beh
• amenazar → ah-meh-nah-THAR
• al vecino → al beh-THEE-no

A few Spain-specific notes:

• z in amenazar is pronounced like th in think
• c before i in vecino is also pronounced like th in Spain
• Stress falls on:
  • NA-die
  • DE-be
  • a-me-na-ZAR
  • ve-CI-no

So a rough full pronunciation is:

NA-dye DEH-beh ah-meh-nah-THAR al beh-THEE-no