Minha garganta dói, então vou beber água quente agora.

Because garganta is a feminine noun in Portuguese, so it takes the feminine possessive minha (my).

• minha garganta = my throat
• meu is used with masculine nouns (e.g., meu braço = my arm).

Portuguese commonly expresses pain with the body part as the subject:

• Minha garganta dói. = (Literally) My throat hurts.
  So the verb agrees with garganta (3rd person singular), not with “I”.

Dói is from the verb doer (“to hurt / to ache”). It’s irregular. Common present forms:

• dói = it hurts (most common for pain statements)
• doem = they hurt (plural body parts)
  Example: Meus olhos doem. = My eyes hurt.

Yes, and it’s very common in Brazil. There’s a nuance:

• Minha garganta dói. = My throat hurts (general statement; can sound a bit more “matter-of-fact”).
• Minha garganta está doendo. = My throat is hurting (emphasizes it happening right now / currently).

Here então introduces a result/consequence (“so/therefore”), and Portuguese normally uses a comma before it when it links two independent parts:
Minha garganta dói, então vou beber água quente agora.
It’s similar to English: “My throat hurts, so I’m going to…”

Here então means so / therefore. But it can also mean:

• then (sequence): Primeiro isso, então aquilo. = First this, then that.
• a conversational “so…”: Então… você vem? = So… are you coming?

vou beber is the “near future” in Portuguese: ir (to go) + infinitive. It’s extremely common in Brazilian Portuguese.

• vou beber = I’m going to drink / I will drink (very natural)
  You can also use the simple future beberei, but it’s less common in everyday speech.

Yes. beber água quente is natural and means “drink hot/warm water.” In real-life contexts Brazilians might also say água morna (warm water), especially for throat discomfort:

• vou beber água morna = I’m going to drink warm water.

quente can cover both, depending on context. If you want to clearly say “warm” (not hot), morno/morna is more specific:

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

In Portuguese, adjectives usually come after the noun:

• água quente (noun + adjective)
  Putting the adjective before (quente água) is generally not normal here.

quente is an adjective that doesn’t change form for masculine vs. feminine (it’s “invariable by gender” in that sense). It only changes for number:

• água quente (singular)
• águas quentes (plural)

agora means now/right now and adds immediacy. vou beber can be “I’m going to drink” soon, but agora makes it explicit that the action is immediate:

• vou beber água quente = I’m going to drink hot water (at some point soon)
• vou beber água quente agora = I’m going to drink hot water now

Yes—Portuguese often drops subject pronouns when the verb ending already shows who it is.
vou clearly indicates “I,” so eu is optional:

• ... então vou beber... (very natural)
• ... então eu vou beber... (also correct; a bit more explicit/emphatic)

You can also hear:

• Tá doendo a minha garganta. (common, informal)
  And in other pain contexts Portuguese often uses an indirect object pronoun:
• Dói-me a cabeça. (more formal / European-leaning style)
  In Brazil, the most common everyday options are Minha garganta dói and Minha garganta tá doendo.

garganta is the standard word for “throat.”
gorja exists but is less common and can sound more specific/colloquial or even regional; learners should stick with garganta in most situations.

A very natural spoken version would be:
Minha garganta tá doendo, então vou beber uma água morna agora.
Notes:

• tá = informal está
• uma água is a common conversational way to refer to “some water / a glass of water.”