Die kalte Luft macht meine Nase rot.

Because Luft is a feminine noun in German.

• die = the feminine singular definite article here
• Luft = air

In this sentence, die kalte Luft is the subject, so it is in the nominative case.

This is because the adjective comes after a definite article and before a noun:

• die kalte Luft

After der / die / das in the nominative singular, adjectives usually take the weak ending -e.

So:

• die kalte Luft
• der kalte Wind
• das kalte Wasser

Here, machen means to make in the sense of to cause something to become something.

The pattern is:

• machen + object + adjective

So:

• Die kalte Luft macht meine Nase rot.
• literally: The cold air makes my nose red.

Other common examples:

• Das macht mich müde. = That makes me tired.
• Die Sonne macht die Haut braun. = The sun makes the skin brown/tan.

It is in the accusative case because it is the direct object of macht.

The cold air is doing something to my nose, so meine Nase is the thing affected.

Even though it is accusative, the form looks the same as nominative because feminine singular nouns often do that:

• nominative: meine Nase
• accusative: meine Nase

Because Nase is feminine.

For a feminine noun in the accusative singular, the possessive stays meine:

• meine Nase
• meine Hand
• meine Jacke

Meinen is used for masculine accusative singular, for example:

• meinen Bruder
• meinen Hund

So meinen Nase would be incorrect.

Because rot is a predicate adjective, not an adjective directly attached to a noun.

Compare:

• eine rote Nase = a red nose
  
  • here the adjective comes before the noun, so it takes an ending
• Meine Nase ist rot. = My nose is red
  
  • here the adjective comes after the verb, so it has no ending
• Die kalte Luft macht meine Nase rot.
  • same idea: rot is describing the result, so it has no ending

So adjectives after verbs like sein, werden, or this kind of machen normally do not take adjective endings.

Because the structure here is:

• subject + verb + object + result adjective

So:

• Die kalte Luft = subject
• macht = verb
• meine Nase = object
• rot = result adjective

German often places this kind of adjective after the object:

• Das macht mich glücklich.
• Der Regen macht die Straße nass.
• Die kalte Luft macht meine Nase rot.

Yes. That is also grammatical, and many learners will hear this kind of structure often in German.

With body parts, German frequently uses a dative pronoun instead of a possessive:

• Die kalte Luft macht mir die Nase rot.
• literally something like: The cold air makes the nose red on me

This often sounds very natural in German.

So both are possible:

• ... macht meine Nase rot
• ... macht mir die Nase rot

The version with mir is often more idiomatic when talking about parts of your own body.

Because all nouns are capitalized in German.

So in this sentence:

• Luft is a noun
• Nase is a noun

That is why they begin with capital letters. This is a standard rule in German spelling.

Yes. Luft here is singular and works like a mass noun, similar to air in English.

So Die kalte Luft means the cold air, not the cold airs.

German can have a plural Lüfte, but that is much less common and usually more literary or poetic. In everyday speech, Luft is usually treated as a singular mass noun.

Yes. It is very normal main-clause word order.

German main clauses usually put the finite verb in second position:

• Die kalte Luft | macht | meine Nase rot

So the sentence is built very naturally:

1. subject
2. finite verb
3. the rest of the sentence

That makes this a good example of standard everyday German word order.