Mi hermano se negó a salir sin paraguas y luego tuvo que disculparse por llegar tarde.

Because the verb here is negarse, not just negar.

• negar = to deny something
  
  • Negó la acusación = He denied the accusation.
• negarse a + infinitive = to refuse to do something
  
  • Se negó a salir = He refused to go out.

So the se is part of the verb’s usual form when it means to refuse.

Because negarse is normally followed by a + infinitive.

This is just the pattern the verb takes:

• negarse a hacer algo = to refuse to do something
• Se negó a hablar = He refused to speak.
• Se negó a salir = He refused to go out.

English learners often want to translate word-for-word, but in Spanish many verbs require a specific preposition before another verb.

After a, when you are using a second verb like this, Spanish uses the infinitive.

So:

• se negó a salir = refused to go out
• not se negó a salió

This is the same idea as:

• empezó a correr = he started to run
• aprendió a cocinar = he learned to cook

When one conjugated verb is followed by another action, the second verb is often left in the infinitive.

In Spanish, after sin, it is very common to leave out the article when speaking generally.

• sin paraguas = without an umbrella
• sin dinero = without money
• sin problemas = without problems

Sin un paraguas is possible, but it sounds more specific or emphatic, as if you mean without even one umbrella or without a particular umbrella. In this sentence, sin paraguas is the most natural choice.

Here it is singular.

Paraguas is one of those Spanish nouns that ends in -s even in the singular.

• el paraguas = the umbrella
• los paraguas = the umbrellas

The article tells you whether it is singular or plural.

So in sin paraguas, it means without an umbrella, not without umbrellas, unless the context clearly suggests plural.

Yes, luego and después can both mean then / afterwards / later in this kind of sentence.

• ...y luego tuvo que disculparse...
• ...y después tuvo que disculparse...

Both are natural. Luego is very common in Spain. In this sentence, it simply links the next event: first he refused to go out without an umbrella, and then later he had to apologize.

Because tuvo que is the preterite, which usually presents the need as a completed event in the story.

• tuvo que disculparse = he had to apologize
• tenía que disculparse = he was supposed to apologize / he needed to apologize / he had to apologize (background or ongoing sense)

Here the sentence tells a sequence of completed actions:

1. He refused to go out.
2. Then he had to apologize.

That makes tuvo que the natural choice.

Because disculparse means to apologize, while disculpar usually means to excuse someone.

Compare:

• Tuvo que disculparse = He had to apologize.
• Tuvo que disculpar a su amigo = He had to excuse his friend.

So the reflexive form changes the meaning.

A very common alternative is:

• tuvo que pedir disculpas = he had to apologize / he had to say sorry

Both are good, though pedir disculpas is often especially common in everyday speech.

Because por here expresses the reason for the apology.

• disculparse por + infinitive/noun = to apologize for something
• por llegar tarde = for arriving late

Para usually expresses purpose or goal, so para llegar tarde would sound like in order to arrive late, which is not the meaning here.

So:

• Se disculpó por llegar tarde = He apologized for arriving late.

Because in Spanish, the natural expression is llegar tarde = to arrive late.

• Llegué tarde = I arrived late / I was late.
• Siempre llega tarde = He always arrives late.

Spanish normally does not say estar tarde for this meaning. The idea is tied to the act of arriving.

Because the sentence tells a completed sequence of past events.

• se negó = he refused
• tuvo que = he had to

These are both in the preterite, which is commonly used for finished actions in a narrative.

The infinitives stay as infinitives because they depend on the main verbs:

• a salir
• disculparse
• llegar

So the structure is:

• conjugated past verb + infinitive
• conjugated past verb + infinitive + por
  • infinitive

Yes, but it is not necessary.

Spanish often leaves out subject pronouns because the verb ending and context already make the subject clear.

• Mi hermano se negó... is enough.
• Él se negó... is also possible, but usually only if you want emphasis or contrast.

Since Mi hermano is already stated, adding él would usually sound redundant.

Yes, it is natural and correct in Spanish from Spain.

A native speaker might also say:

• Mi hermano se negó a salir sin paraguas y luego tuvo que pedir disculpas por llegar tarde.
• Mi hermano no quiso salir sin paraguas y luego tuvo que disculparse por llegar tarde.

But your original sentence is completely natural. It uses standard structures and vocabulary that are very common in Spain.