На балконі вже сухо після дощу.

Because на балконі expresses location: the action/state is happening on the balcony.

With на, Ukrainian often uses:

• Accusative for direction/movement toward a place: йти на балкон = to go out onto the balcony
• Locative for being in/at/on a place: бути на балконі = to be on the balcony

In this sentence, nothing is moving. It describes a state, so балконі is in the locative case.

Балкон is a masculine noun. After the preposition на when it means location, it takes the locative singular form.

So:

• балкон = balcony
• на балконі = on the balcony

This -і ending is a normal locative ending for many nouns.

In the present tense, Ukrainian usually omits the verb to be.

So:

• На балконі вже сухо literally looks like On the balcony already dry
• but it means It is already dry on the balcony

This is completely normal in Ukrainian.
If you wanted past or future, a form of бути would appear:

• На балконі було сухо = It was dry on the balcony
• На балконі буде сухо = It will be dry on the balcony

Сухо here is not an adjective agreeing with a noun. It is a predicative adverb / category of state word meaning it is dry.

Ukrainian often uses words like this to describe a general condition of a place or environment:

• тепло = it is warm
• холодно = it is cold
• темно = it is dark
• сухо = it is dry

So На балконі вже сухо means The balcony area is already dry / It is already dry on the balcony, not the balcony is a dry one in an adjective-agreement sense.

Вже means already.

It shows that the state has changed or has been reached by now:

• before, it was still wet
• now, it is already dry

In this sentence, вже adds the idea of now / by this point.

The preposition після means after and it requires the genitive case.

So:

• дощ = rain
• після дощу = after the rain

Here дощу is the genitive singular form of дощ.

Yes. Ukrainian word order is fairly flexible, and changing it often changes emphasis, not basic meaning.

For example:

• На балконі вже сухо після дощу. = neutral
• Після дощу на балконі вже сухо. = emphasizes after the rain
• Вже сухо на балконі після дощу. = emphasizes already dry

The original version sounds natural and balanced.

Yes. Ukrainian normally says на балконі, just as English says on the balcony.

This is simply the standard preposition used with балкон:

• вийти на балкон = go out onto the balcony
• стояти на балконі = stand on the balcony

Using в балконі would not be normal here.

Yes, it is essentially an impersonal sentence.

There is no explicit grammatical subject like it. English needs it is dry, but Ukrainian often does not use a subject in this kind of sentence:

• сухо = it is dry
• тихо = it is quiet
• спекотно = it is hot

So the sentence describes the condition of the place without naming a subject.

Yes, very naturally:

• Після дощу на балконі вже сухо.

This version may sound slightly more natural if you want to highlight the time relationship first: after the rain. Both versions are correct.

In practice, it describes the condition there—the balcony space/surface/environment.

So the sentence can be understood as:

• It’s already dry on the balcony
• The balcony is already dry after the rain

Ukrainian often leaves this broad and natural, without specifying whether you mean the floor, the air, or the general area. Context usually makes it clear.