После дождя свежесть воздуха в парке радует меня.

Because после requires the genitive case.

• дождь = nominative singular, the dictionary form
• дождя = genitive singular

So:

• после дождя = after the rain

This is something you simply learn with the preposition:

• после урока = after the lesson
• после работы = after work
• после фильма = after the movie

So in this sentence, дождя is in the form demanded by после.

The subject is свежесть воздуха в парке.

More specifically, the core subject is свежесть (freshness), and the rest describes it:

• свежесть = freshness
• воздуха = of the air
• в парке = in the park

So the sentence is structured like:

• [После дождя] [свежесть воздуха в парке] [радует] [меня].

Literally, it works like:

• After the rain, the freshness of the air in the park pleases me.

Because it is the subject of the sentence.

In Russian, the thing doing the action or being described is usually in the nominative. Here, freshness is what pleases me, so свежесть is nominative.

You can think of it this way:

• свежесть радует = the freshness pleases

That is why свежесть stays in its basic dictionary form.

Because воздуха depends on свежесть and means of the air.

Russian often uses the genitive case after one noun to define another noun:

• свежесть воздуха = freshness of the air
• запах кофе = smell of coffee
• цвет неба = color of the sky

So:

• воздух = air
• воздуха = of the air

This is a very common noun + genitive pattern in Russian.

Because в meaning in takes the prepositional case when talking about location.

• парк = park
• в парке = in the park

So в парке tells you where this fresh air is being experienced.

Compare:

• в доме = in the house
• в школе = at school / in the school
• в городе = in the city

If you are talking about motion into a place, Russian often uses в + accusative instead, but here it is just location, so в парке is prepositional.

Because the subject, свежесть, is singular.

A useful thing to notice is that in the present tense, Russian third-person singular verbs do not show gender. So:

• он радует
• она радует
• оно радует

All are радует.

Since свежесть is singular, the verb must also be singular:

• свежесть радует

If the subject were plural, you would get:

• запахи радуют меня = the smells please me

Because радовать takes a direct object, so the person affected goes in the accusative case.

• радовать кого? = to please whom?
• меня = me (accusative)

So:

• свежесть воздуха радует меня = the freshness of the air pleases me

This is different from verbs like нравиться, which use мне:

• мне нравится свежесть воздуха = I like the freshness of the air

Both are natural, but the grammar is different:

• радовать + accusative → меня
• нравиться + dative → мне

Yes, absolutely.

A very natural alternative is:

• После дождя мне нравится свежесть воздуха в парке.

Both sentences are correct, but they feel slightly different:

• свежесть воздуха в парке радует меня
  = more literary or descriptive, literally the freshness ... pleases me
• мне нравится свежесть воздуха в парке
  = more everyday, literally I like the freshness ...

So if you are speaking casually, мне нравится may feel more common. The original sentence sounds a bit more polished or expressive.

No, Russian word order is fairly flexible.

The original sentence:

• После дождя свежесть воздуха в парке радует меня.

is completely natural. But you could also say:

• Свежесть воздуха в парке после дождя радует меня.
• Меня радует свежесть воздуха в парке после дождя.
• После дождя меня радует свежесть воздуха в парке.

These versions all mean roughly the same thing, but the focus changes.

For example:

• После дождя at the beginning sets the scene first: After the rain...
• Меня радует earlier can put more emphasis on my reaction
• putting свежесть воздуха earlier highlights the thing being talked about

So the original word order is natural, but not the only possible one.

In practice, it most naturally describes the air/freshness being talked about: the freshness of the air in the park.

So most learners should understand it as:

• свежесть [воздуха в парке]

However, Russian word order can allow some overlap in interpretation, and listeners often understand it from context as part of the whole scene after the rain.

If you wanted to make the location feel even more clearly attached to the whole sentence, you might move it:

• После дождя в парке свежесть воздуха радует меня.

If you wanted to keep it clearly close to воздуха, the original already does that quite well.

A careful stress pattern is:

• По́сле дождя́ све́жесть во́здуха в па́рке ра́дует меня́.

A few important stresses:

• по́сле
• дождя́
• све́жесть
• во́здуха
• па́рке
• ра́дует
• меня́

For an English speaker, one useful point is that Russian stress is mobile and must often be learned word by word. In this sentence, the stress helps distinguish the natural rhythm of the phrase.

It is natural Russian.

It sounds a little more written or descriptive than very casual speech, but it is absolutely normal and correct. A native speaker could use it in speech, especially in a thoughtful or expressive context.

If you wanted a more conversational version, you might hear something like:

• После дождя мне нравится свежий воздух в парке.
• После дождя меня радует свежий воздух в парке.

But the original sentence is perfectly good Russian and has a nice, slightly literary tone.