O meu nariz fica vermelho quando faz frio e eu espirro constantemente.

In European Portuguese, possessive adjectives almost always come with a definite article:

• o meu nariz – my nose
• a minha casa – my house
• os meus livros – my books

So o meu nariz is the normal, standard way to say my nose in Portugal.

You’ll typically omit the article mainly in a few special cases, for example:

• After the verb ser: É meu. – It’s mine.
• In some vocative or very informal/poetic styles.

But as a learner of Portuguese from Portugal, a good rule is: use the article with possessives before a noun.

People will understand you, but in European Portuguese it sounds either:

• Brazilian, or
• poetic / very marked, not neutral everyday speech.

In normal Portugal usage you would say:

• O meu nariz fica vermelho.

In Brazilian Portuguese, Meu nariz fica vermelho is much more natural and common, but the learner here is focusing on Portugal, so stick with o meu.

Ficar is a very common verb with several uses. Here it is a linking verb meaning:

• to become / to get / to turn.

So fica vermelho = gets red / turns red.

Compare:

• O meu nariz é vermelho. – My nose is red (as a permanent characteristic – strange unless you’re a clown).
• O meu nariz está vermelho. – My nose is red (right now) – a temporary state.
• O meu nariz fica vermelho. – My nose gets / becomes red (whenever it’s cold – a change of state, habitually).

So fica is the best choice for “gets” or “turns” red in repeated situations.

Portuguese often uses fazer impersonal­ly to talk about weather conditions, similar to French il fait froid:

• Faz frio. – It’s cold.
• Faz calor. – It’s hot.
• Faz sol. – It’s sunny.
• Faz vento. – It’s windy.

There is no subject pronoun in Portuguese here — you don’t say *ele faz frio.

So quando faz frio literally means “when it makes cold”, but idiomatically: “when it’s cold”.

Both are very common and both can mean “it’s cold”:

• Faz frio.
• Está frio.

In everyday speech in Portugal:

• Está frio is maybe slightly more common in casual conversation about the current temperature.
• Faz frio is also perfectly natural, often used in neutral descriptions of weather.

For your sentence, you could also say:

• O meu nariz fica vermelho quando está frio…

No real change in meaning; both sound fine to a European Portuguese speaker.

Portuguese uses the present indicative for:

1. Things that are happening now, and
2. Habits / general truths / repeated situations.

Your sentence is about a habitual reaction:

• O meu nariz fica vermelho quando faz frio e eu espirro constantemente.
  – My nose gets red when it’s cold and I sneeze constantly (whenever that situation happens).

So the simple present is exactly right, just like in English “When it’s cold, my nose gets red and I sneeze a lot.”

Both are possible, but they mean slightly different things:

• Quando faz frio, o meu nariz fica vermelho.
  – General, habitual fact: whenever it’s cold in general, my nose gets red.

• Quando fizer frio, o meu nariz vai ficar vermelho.
  – Refers more to a specific future time: when it gets cold (in the future), my nose will get red.

In your sentence, you’re describing a general tendency, so the present indicative (faz) is the natural choice in European Portuguese.

Yes, you absolutely can drop eu:

• …e eu espirro constantemente.
• …e espirro constantemente.

Both are grammatically correct.

Because Portuguese verb endings already show the person (espirro clearly means I sneeze), the subject pronoun is often optional.

Using eu gives a bit more emphasis or contrast, for example:

• O meu nariz fica vermelho quando faz frio e eu espirro constantemente.
  – My nose gets red and I sneeze constantly (maybe as opposed to other people, or emphasising yourself).

The most natural position is after the verb:

• Eu espirro constantemente. – I sneeze constantly.

Other possibilities:

• Constantemente eu espirro. – Possible, but sounds more literary/marked.
• Eu constantemente espirro. – Understandable, but not the most natural rhythm in everyday speech.

General rule for adverbs like constantemente, sempre (always):

• Default: verb + adverb
  
  • espirro sempre, espirro muito, espirro constantemente.

Here, espirro is a verb form:

• It is the 1st person singular, present indicative of espirrar (to sneeze).

Full present conjugation of espirrar:

• eu espirro – I sneeze
• tu espirras – you sneeze (informal singular)
• ele / ela / você espirra – he/she/you sneeze(s)
• nós espirramos – we sneeze
• vocês espirram – you (plural) sneeze
• eles / elas espirram – they sneeze

There is also a noun um espirro = a sneeze, but in your sentence it’s the verb.

Nariz is a masculine noun:

• o nariz – the nose

Adjectives must agree in gender and number with the noun:

• o nariz vermelho – the red nose (masculine singular)
• os narizes vermelhos – the red noses (masculine plural)
• a cara vermelha – the red face (feminine singular)
• as caras vermelhas – the red faces (feminine plural)

So we say:

• O meu nariz fica vermelho.
  – vermelho matches nariz (masculine singular).

Your sentence is correct and natural. A European Portuguese speaker might also say, for example:

• O meu nariz fica vermelho quando está frio e estou sempre a espirrar.
• No inverno o meu nariz fica vermelho e passo a vida a espirrar.
  (literally “I spend my life sneezing” = I sneeze all the time)

But:

• O meu nariz fica vermelho quando faz frio e eu espirro constantemente.

is perfectly fine, idiomatic Portuguese from Portugal.