У мене нежить, і мій ніс червоний.

Because Ukrainian very often expresses possession with у/в + genitive pronoun/noun, literally something like at me.

So:

• У мене нежить = literally At me, [there is] a runny nose
• Natural English translation: I have a runny nose

This pattern is especially common for:

• possessions: У мене є машина = I have a car
• symptoms/states: У мене кашель = I have a cough
• age: Мені 20 років uses a different pattern, but also not I have

You can sometimes use мати = to have, but with symptoms like this, У мене нежить is much more natural.

Мене is the genitive form of я.

The pattern у мене requires the genitive:

• я = I
• мене = of me / me
• у мене = at me / with me

So in У мене нежить, мене is not the subject. It is part of the possession construction.

In the present tense, Ukrainian often omits the verb є (is/are) in simple statements.

So:

• У мене нежить literally has no spoken is, but it is a complete sentence
• Мій ніс червоний = My nose is red, again without saying is

This is very normal in Ukrainian.

Compare:

• Мій ніс червоний = My nose is red
• Вона лікар = She is a doctor

You usually do not say мій ніс є червоний in ordinary speech.

Нежить means a runny nose, and depending on context it can also suggest nasal congestion/cold symptoms.

It refers to the symptom itself, not necessarily a full cold.

So:

• У мене нежить = I have a runny nose
• It does not mean I am sick in a general sense
• It is more specific than I have a cold

Not always.

In English, body parts usually take a possessive:

• my nose
• my hand
• my head

In Ukrainian, the possessive is often omitted if the owner is obvious.

So these are all possible:

• Мій ніс червоний = My nose is red
• У мене червоний ніс = I have a red nose
• Ніс у мене червоний = My nose is red / As for my nose, it’s red

Your sentence is correct, but У мене червоний ніс may sound a bit more natural in everyday speech.

Because червоний has to agree with ніс.

Ніс is:

• masculine
• singular
• nominative

So the adjective must also be masculine singular nominative:

• червоний ніс = red nose

Compare:

• червона рука = red hand
• червоне око = red eye
• червоні губи = red lips

Because і is joining two separate clauses:

1. У мене нежить
2. мій ніс червоний

Each part has its own complete idea, so Ukrainian normally uses a comma before і here.

Compare:

• У мене нежить, і мій ніс червоний. = comma needed
• Мій ніс червоний і болить. = usually no comma, because this is one subject with two predicates

Yes. У мене and В мене mean the same thing here.

Ukrainian often alternates у and в for smoother pronunciation. This is called euphony.

So both are correct:

• У мене нежить
• В мене нежить

At the beginning of a sentence, both are common. Native speakers often choose whichever sounds better with the surrounding words.

The word order is fairly flexible.

Your sentence is normal:

• У мене нежить, і мій ніс червоний.

But Ukrainian could also say:

• В мене нежить, і ніс у мене червоний.
• У мене нежить, і у мене червоний ніс.
• Мій ніс червоний, і в мене нежить.

The meaning stays similar, but the emphasis can shift a little.

A useful thing to remember:

• English often has more fixed word order
• Ukrainian uses word order more freely for emphasis, rhythm, and style

A helpful stressed version is:

У мене́ не́жить, і мій ніс черво́ний.

Rough pronunciation:

• У мене ≈ oo meh-NEH
• нежить ≈ NEH-zhyt
• і ≈ ee
• мій ≈ meey
• ніс ≈ nees
• червоний ≈ cher-VOH-nyy

A few sound notes:

• ж sounds like the s in measure
• ч sounds like ch in church
• и is not exactly English ee; it is a shorter, more central vowel
• й sounds like y in boy or yes, depending on position

Yes. Your sentence is correct, but a very natural alternative would be:

У мене нежить, і в мене червоний ніс.

Or even more compactly:

У мене нежить і червоний ніс.

That said, У мене нежить, і мій ніс червоний is still grammatical and understandable. A learner should mainly notice:

• У мене for I have
• no present-tense is
• adjective agreement in червоний