Po běhání mi nevadí chladný vzduch.

Here po means after, and with that meaning it normally takes the locative case.

• po běhání = after running
• po obědě = after lunch
• po práci = after work

The noun běhání is a verbal noun meaning running as an activity. In this sentence, its locative form looks the same as its dictionary form, so you do not see a visible ending change.

So:

• po + locative = after

Běhání means running in the sense of the activity or pastime.

Compare:

• běhání = running, the activity
• běh = a run, a race, a single run

So:

• Po běhání mi nevadí chladný vzduch.
  = After running / After I’ve been running, cold air doesn’t bother me.

• Po běhu mi nevadí chladný vzduch.
  = After the run, cold air doesn’t bother me.

Both can work, but běhání focuses more on the activity itself, while běh sounds more like a specific run or event.

Mi is the short dative form of já and means to me.

The verb vadit works differently from English to bother. In Czech, the person who is bothered is usually in the dative:

• Vadí mi hluk. = Noise bothers me.
• Nevadí mi zima. = Cold doesn’t bother me.

So in this sentence:

• mi = to me
• literally: After running, cold air does not bother to me
• natural English: After running, cold air doesn’t bother me

Because vadit takes the dative, not the accusative.

Forms of I / me here:

• mě = accusative/genitive
• mně / mi = dative

Since vadit uses the dative for the affected person, you need:

• vadí mi
• not vadí mě

Also, mi is the short unstressed form, which is very common in ordinary sentences.

You could also say:

• Po běhání mně nevadí chladný vzduch.

That is grammatically correct, but mi is more usual here.

The verb agrees with the grammatical subject, which is chladný vzduch.

So the structure is:

• chladný vzduch = subject, singular
• nevadí = 3rd person singular present

This is important because English learners often feel that mi should control the verb, but it does not. Mi is only the dative object.

Think of it like this:

• Chladný vzduch nevadí. = Cold air is not bothersome.
• Chladný vzduch mi nevadí. = Cold air is not bothersome to me.

If the subject were plural, the verb would change:

• Chladné dny mi nevadí. = Cold days don’t bother me.

In the present tense, singular and plural of vadit can both look like vadí, so context matters.

Because chladný vzduch is the subject of the sentence, so it is in the nominative.

• chladný = cold
• vzduch = air

Together:

• chladný vzduch = cold air

The sentence is built around the idea:

• Cold air doesn’t bother me.

So cold air is the thing doing the “bothering” grammatically, even though in English the structure may feel a little different.

Usually, yes, but it can cover a range of meanings depending on context:

• to bother
• to annoy
• to be a problem
• to mind

For example:

• Vadí mi hluk. = Noise bothers me / I mind the noise.
• Nevadí ti to? = Does that bother you? / Do you mind?
• To nevadí. = That’s okay / Never mind / It doesn’t matter.

So in your sentence, nevadí is best understood as doesn’t bother or I don’t mind.

Yes, Czech word order is flexible. The version you have is natural and emphasizes the setting first:

• Po běhání = after running
• mi = to me
• nevadí chladný vzduch = cold air doesn’t bother me

Other possible orders:

• Chladný vzduch mi po běhání nevadí.
• Po běhání chladný vzduch mi nevadí.
  This is possible, but less neutral.
• Mně po běhání chladný vzduch nevadí.
  More emphasis on me.

The original sentence sounds neutral and natural because it starts with the time/context phrase and then places the short clitic mi early, which is typical in Czech.

Because mi is a clitic: a short unstressed word that tends to appear near the beginning of the clause, usually in the second position.

In:

• Po běhání mi nevadí chladný vzduch.

the first unit is Po běhání, and then the clitic mi comes right after it.

This is a very common Czech pattern. Similar examples:

• Po práci mi chutná pivo.
• V zimě mi vadí tma.
• Dnes mi není dobře.

So the position of mi is not random; it follows a common word-order rule for short unstressed forms.

Not exactly. Chladný vzduch literally means cold air.

If you wanted cold weather, you would usually say something like:

• chladné počasí = cold weather

However, in real life, chladný vzduch can imply being outside in cool air, so the practical meaning may overlap a little with English expressions about cold conditions. Still, grammatically and literally, it is cold air.

Yes, that is possible, but chladný and studený are not exactly identical.

• chladný = cool, chilly, cold in a milder or more neutral sense
• studený = cold, often more strongly cold

So:

• chladný vzduch sounds natural for cool or chilly air
• studený vzduch can sound colder or harsher

In many contexts both are possible, but chladný vzduch is a very natural choice here.

By itself, it can be understood as either, depending on context.

It can mean:

• a general statement: After running, cold air doesn’t bother me
• a specific situation: After this run / after I’ve been running, cold air doesn’t bother me

Czech present tense often covers both general and immediate meanings when the context is clear.

If you wanted to make it more clearly habitual, context would usually do that:

• Po běhání mi obvykle nevadí chladný vzduch.
  = After running, cold air usually doesn’t bother me.

If you wanted a clearly one-time past meaning, you would change the tense:

• Po běhání mi nevadil chladný vzduch.
  = After running, cold air didn’t bother me.