Quando estou com dor de garganta, eu prefiro beber água quente com mel.

In Brazilian Portuguese, estar com is a very common way to talk about temporary physical conditions or symptoms.

So:

• estou com dor de garganta = I have a sore throat
• literally, it is closer to I am with a sore throat

This pattern is extremely natural in everyday speech:

• Estou com febre = I have a fever
• Estou com dor de cabeça = I have a headache
• Estou com gripe = I have the flu

You may also hear tenho dor de garganta, but estou com dor de garganta sounds very natural for a current condition.

Literally, dor de garganta is pain of throat or throat pain.

Portuguese often uses noun + de + noun to describe a type of pain:

• dor de cabeça = headache
• dor de dente = toothache
• dor de garganta = sore throat / throat pain

The de here works a bit like of in English, but the full expression is best learned as a chunk: dor de garganta.

Yes, you can hear both, but they are not always used in exactly the same way.

• dor de garganta is the standard set expression for sore throat
• dor na garganta means pain in the throat and can sound a little more literal or specific

So:

• Estou com dor de garganta = I have a sore throat
• Estou com dor na garganta = I have pain in my throat

In many situations, the meaning overlaps, but dor de garganta is the more common everyday phrase for the symptom.

That is true: Portuguese often drops subject pronouns because the verb ending already shows the person.

So both are correct:

• Quando estou com dor de garganta, prefiro beber água quente com mel.
• Quando estou com dor de garganta, eu prefiro beber água quente com mel.

Adding eu can:

• give a little emphasis
• make the sentence clearer
• sound more natural in some spoken contexts

In Brazilian Portuguese, subject pronouns are used more often than in European Portuguese, so eu prefiro sounds completely normal.

Both are possible, but they mean slightly different things.

• Quando estou com dor de garganta... = whenever I have a sore throat / when I am experiencing it
  This sounds habitual or general.
• Quando eu estiver com dor de garganta... = when I am having a sore throat / when I happen to have one in the future
  This points more to a future situation.

In your sentence, the speaker is talking about a usual preference, so Quando estou... is a very natural choice.

In Portuguese, adjectives usually come after the noun.

So:

• água quente = hot water
• chá quente = hot tea
• mel puro = pure honey

Putting the adjective before the noun is much less common and often changes the tone or meaning. For ordinary description, noun + adjective is the normal pattern.

Yes, you absolutely could.

Both are natural:

• beber água quente com mel
• tomar água quente com mel

In Brazilian Portuguese:

• beber is specifically to drink
• tomar is also often used for drinking, especially in everyday speech

Examples:

• Vou beber água.
• Vou tomar água.

Both work, though beber is a little more literal. In casual Brazilian speech, tomar is very common for drinks.

Yes. com mel means with honey.

So água quente com mel means:

• hot water with honey
• honey added to hot water

This is a very common structure in Portuguese:

• café com leite = coffee with milk
• pão com manteiga = bread with butter
• chá com limão = tea with lemon

The sentence is natural, but what people actually say depends on what they mean.

• água quente com mel = hot water with honey
• chá com mel = tea with honey

Both are normal. If the person specifically means plain hot water plus honey, then água quente com mel is correct. If they mean tea, they would usually say chá com mel.

So the sentence is natural as written; it is just specific.

Here are a few pronunciation notes for Brazilian Portuguese:

• Quando → roughly KWAN-doo
• estou → roughly es-TOH
• dor → roughly dohr
• garganta → roughly gar-GAN-ta
• prefiro → roughly preh-FEE-roo
• beber → roughly beh-BEHR
• água → roughly AH-gwah
• quente → roughly KEN-chee or KEN-tee, depending on accent
• mel → roughly meh-o or mel, depending on region and speed

A couple of useful points:

• qu in quando and quente sounds like kw
• gua in água is pronounced as part of one flowing syllable group
• final r in Brazilian Portuguese is often softer than in English, and in some accents it becomes almost like h

Portuguese does not always use articles in the same places English does.

In this sentence:

• dor de garganta is a fixed expression, so no article is needed
• mel is being used as a substance in a general sense, so com mel is normal

Compare:

• Estou com dor de garganta. = I have a sore throat
• Eu comprei o mel. = I bought the honey

When you mean honey in general as an ingredient, com mel is the natural form.

It sounds like a general habit or preference.

Because of Quando estou... eu prefiro..., the meaning is something like:

• whenever I have a sore throat, I prefer...
• when I get a sore throat, I prefer...

So the speaker is not just describing one moment right now. They are saying what they usually prefer in that situation.

Yes, a Brazilian could say this. It is grammatical and natural.

A few everyday alternatives might be:

• Quando estou com dor de garganta, prefiro tomar água quente com mel.
• Quando fico com dor de garganta, prefiro tomar água morna com mel.
• Pra dor de garganta, eu prefiro água morna com mel.

A small note:

• água quente = hot water
• água morna = warm/lukewarm water

In real life, many people might prefer morna rather than quente, depending on what they actually mean.