De kou maakt mijn wang rood, en mijn lip voelt droog aan.

Kou is the noun cold. So de kou means the cold or the cold weather.

Dutch often uses a noun here where English might also use an adjective:

• Het is koud. = It is cold.
• De kou maakt mijn wang rood. = The cold makes my cheek red.

The article de is there because kou is a common-gender noun: de kou.

Because the subject is de kou, which is singular.

The verb maken is the infinitive. In the present tense:

• ik maak
• jij maakt
• hij/zij/het maakt

So:

• De kou maakt ... = The cold makes ...

The same thing happens later in the sentence:

• mijn lip voelt ...

because mijn lip is also singular.

Here maken means to make or to cause to become.

The pattern is:

iets/iemand + adjective + maken

So:

• De kou maakt mijn wang rood. = The cold makes my cheek red. = The cold turns my cheek red.

Other examples:

• Dat maakt me blij. = That makes me happy.
• De zon maakt de kamer warm. = The sun makes the room warm.

So rood maken is basically to make red / to turn red.

Because rood is being used as a predicative adjective, not directly before a noun.

Compare:

• de rode wang = the red cheek
  Here the adjective comes before the noun, so it usually takes -e.
• mijn wang is rood = my cheek is red
• de kou maakt mijn wang rood = the cold makes my cheek red

After verbs like zijn, worden, blijven, maken, and aanvoelen, adjectives usually stay in the basic form:

• rood
• droog
• warm
• zacht

Because mijn already functions as a determiner, so you do not also use de or het.

So you say:

• mijn wang = my cheek
• mijn lip = my lip

Not:

• de mijn wang
• de mijn lip

This works the same way as in English: you say my cheek, not the my cheek.

Wang means cheek, singular. So this sentence is talking about one cheek.

• mijn wang = my cheek
• mijn wangen = my cheeks

If you meant both cheeks, you would say:

• De kou maakt mijn wangen rood.

So the singular here is completely grammatical; it just refers to one cheek.

Here mijn lip means my lip, singular, so it refers to one lip.

If you want to talk about both lips, Dutch would normally use the plural:

• Mijn lippen voelen droog aan. = My lips feel dry.

So the sentence as written is specifically singular. It is not a special Dutch rule forcing the singular; it simply means one lip.

This comes from the separable verb aanvoelen, which means to feel or to feel like in the sense of sensation or texture.

So:

• Mijn lip voelt droog aan. = My lip feels dry. = My lip feels dry to the touch / in sensation.

This is very common with adjectives:

• De stof voelt zacht aan. = The fabric feels soft.
• De lucht voelt warm aan. = The air feels warm.
• Zijn huid voelt koud aan. = His skin feels cold.

It is slightly different from simply is droog, because voelt droog aan emphasizes how it feels.

Because aanvoelen is a separable verb.

Its full infinitive is:

• aanvoelen = to feel / to seem by touch

In a main clause, the verb splits:

• Mijn lip voelt droog aan.

So:

• voelt = the conjugated verb part
• aan = the separable particle, placed later in the clause

This happens with many Dutch separable verbs:

• opstaan → Ik sta vroeg op.
• aankomen → De trein komt laat aan.
• aanvoelen → Het voelt vreemd aan.

For the same reason as rood earlier: droog is a predicative adjective.

Compare:

• de droge lip = the dry lip
• mijn lip voelt droog aan = my lip feels dry

When the adjective comes before a noun, it often gets -e:

• de droge huid
• de rode wang

But after verbs like zijn or aanvoelen, it usually stays uninflected:

• De huid is droog.
• Mijn lip voelt droog aan.

Yes. Both are possible, but the nuance is a little different.

• Mijn lip is droog.
  This simply states a fact: my lip is dry.

• Mijn lip voelt droog aan.
  This focuses more on the sensation or how it seems to the touch.

In everyday speech, both can work. Voelt droog aan is especially natural when describing how something physically feels.

Yes. After en, the second part is just another main clause with normal word order.

• De kou maakt mijn wang rood
• en mijn lip voelt droog aan

In the second clause:

• mijn lip = subject
• voelt = finite verb

That is standard Dutch main-clause word order.

If you break it apart, it is simply:

• Mijn lip voelt droog aan.

So nothing unusual is happening there; en is just linking two main clauses.