Mně nevadí chladný vzduch, ale mojí sestře vadí hlasitá hudba od sousedů.

Because mně is the dative form of já (I), and the verb vadit works with the person affected in the dative.

So Czech says, literally:

Cold air does not bother to me

rather than

I do not mind cold air

With vadit:

• the person who feels bothered = dative
• the thing that is bothersome = nominative

So:

• Mně nevadí chladný vzduch = Cold air doesn’t bother me
• not Já nevadí...

Vadit means something like to bother, to annoy, or in negative sentences often to not mind.

Its pattern is:

something bothers someone

In Czech grammar:

• something = nominative subject
• someone = dative

So in your sentence:

• chladný vzduch is the thing that does not bother
• mně is the person affected
• hlasitá hudba is the thing that does bother
• mojí sestře is the person affected

This is why the structure may feel “backwards” to an English speaker.

Because they are the subjects of the verb vadit.

Even though English often puts the person first (I don’t mind cold air), Czech treats cold air and loud music as the grammatical subjects:

• chladný vzduch nevadí
• hlasitá hudba vadí

That is why both noun phrases are in the nominative:

• chladný vzduch = masculine singular nominative
• hlasitá hudba = feminine singular nominative

Their adjectives also match:

• chladný matches vzduch
• hlasitá matches hudba

Mojí sestře is dative singular and means to my sister or simply my sister in this structure.

It is dative because vadit requires the person affected to be in the dative.

Breakdown:

• mojí = dative feminine singular of moje / má (my)
• sestře = dative singular of sestra (sister)

So:

• mojí sestře vadí... = ...bothers my sister

In Czech, you do not need a separate word like to here, because the dative ending already carries that function.

Yes. Mé sestře and mojí sestře are both correct.

Both mean to my sister. The difference is mainly stylistic:

• mé sestře = shorter form
• mojí sestře = longer form, very common in speech and also standard

So these are both fine:

• Mé sestře vadí hlasitá hudba.
• Mojí sestře vadí hlasitá hudba.

They are usually the same grammatical form: both can mean to me in the dative.

The main difference is style and position:

• mi is the short, unstressed form
• mně is the full, stressed form

In this sentence, Mně is used because it stands at the beginning and gives contrast:

• Mně nevadí... ale mojí sestře vadí...
• As for me, it doesn’t bother me, but my sister is bothered...

A very natural alternative would be:

• Chladný vzduch mi nevadí, ale mojí sestře vadí hlasitá hudba od sousedů.

But mi usually cannot stand in the first position by itself.

Yes, Czech word order is flexible.

The sentence starts with Mně and then mojí sestře to create a clear contrast between the two people:

• Mně nevadí...
• ale mojí sestře vadí...

That gives the sentence a strong me vs. my sister feeling.

Other word orders are possible, for example:

• Chladný vzduch mi nevadí, ale hlasitá hudba od sousedů mojí sestře vadí.
• Mně chladný vzduch nevadí, ale mojí sestře hlasitá hudba od sousedů vadí.

The meaning stays basically the same, but the emphasis changes.

It can be translated in a few ways depending on context:

• vadí = bothers, annoys, is a problem for
• nevadí = doesn’t bother, isn’t a problem, doesn’t mind

So:

• Mně nevadí chladný vzduch = Cold air doesn’t bother me / I don’t mind cold air
• mojí sestře vadí hlasitá hudba = Loud music bothers my sister

It usually does not mean something extremely strong like can’t stand unless the context makes that clear.

Od sousedů means from the neighbors or coming from the neighbors / next door.

The preposition od takes the genitive case, so sousedé changes to sousedů.

So:

• od
  • genitive
• sousedů = genitive plural of soused / sousedé

In this sentence:

• hlasitá hudba od sousedů = loud music from the neighbors

It describes the source of the music.

Because sestra is the nominative form, but after vadit we need the dative for the person affected.

So the noun changes:

• sestra = nominative
• sestře = dative

This kind of change is normal in Czech declension.

A few forms are:

• sestra = sister
• bez sestry = without a sister
• k sestře = to a sister
• vidím sestru = I see a sister

In your sentence, the correct form is sestře because it means to my sister in grammatical terms.

Because Czech adjectives and possessive words must agree with the noun they describe in gender, number, and case.

Here:

• chladný vzduch

  • vzduch is masculine singular nominative
  • so the adjective is chladný

• hlasitá hudba

  • hudba is feminine singular nominative
  • so the adjective is hlasitá

• mojí sestře

  • sestře is feminine singular dative
  • so my becomes mojí

This agreement is one of the most important features of Czech grammar.