После дождя на улице грязно.

Because после always takes the genitive case. The basic dictionary form is дождь (rain), but after после it changes to дождя.

• дождь = nominative
• после дождя = after rain / after the rain

This is a very common pattern in Russian:

• после урока = after the lesson
• после работы = after work
• после обеда = after lunch

So после дождя is just the normal case form required by the preposition.

Because на улице is a fixed, very common Russian expression meaning outside or out on the street.

Here улица changes to the prepositional case because it follows на in a location meaning:

• улица = street
• на улице = on the street / outside

So grammatically, this is a location phrase: where is it dirty? → на улице.

It can mean both, depending on context.

In many everyday sentences, на улице is best understood as outside:

• На улице холодно. = It is cold outside.
• На улице темно. = It is dark outside.

In your sentence, После дождя на улице грязно, it can suggest:

• outside in general, especially around the streets and ground
• literally on the street/outdoors

So English might translate it as outside it’s dirty/muddy after the rain, even though the Russian wording uses street.

Because in the present tense, Russian normally does not use a verb corresponding to is / am / are.

So:

• На улице грязно. literally looks like Outside dirty
• but it means It is dirty outside

This is completely normal Russian grammar. The idea of is is understood automatically.

Compare:

• Он дома. = He is at home.
• Она счастлива. = She is happy.
• Здесь тихо. = It is quiet here.

Here грязно is not functioning like a regular adjective describing a noun. It is being used as a predicative word meaning it is dirty / muddy.

Compare:

• грязная улица = a dirty street
  Here грязная is an adjective modifying улица.
• На улице грязно = It is dirty outside / on the street
  Here грязно describes the situation or environment.

This use is very common in Russian with words like:

• холодно = cold
• тепло = warm
• темно = dark
• тихо = quiet
• чисто = clean
• грязно = dirty

Because the sentence is not saying the street is dirty as a noun + adjective structure. It is saying it is dirty outside as a general condition.

So:

• Улица грязная. = The street is dirty.
  This describes улица directly, so the adjective agrees with it.
• На улице грязно. = It is dirty outside / on the street.
  This describes the environment, so Russian uses грязно.

That is an important distinction:

• грязная = adjective agreeing with a feminine noun
• грязно = impersonal predicative form

Yes, absolutely, but it means something slightly different in emphasis.

• После дождя на улице грязно.
  Focus: the general outdoor condition after the rain
• Улица грязная после дождя.
  Focus: the street itself is dirty after the rain

The first sentence sounds broader and more natural for everyday comments about weather and walking conditions. The second is more specifically about the street as an object.

Yes, Russian word order is flexible, but the original order is very natural:

• После дождя на улице грязно.

This order gives:

1. time/background: after the rain
2. place: outside / on the street
3. main comment: it’s dirty

You could rearrange it, for example:

• На улице грязно после дождя.
• Грязно на улице после дождя.

These are grammatically possible, but they may shift emphasis. The original sentence sounds neutral and natural.

Literally, грязно means dirty. But in weather contexts, especially after rain, it often strongly suggests muddy, messy, or slushy conditions outdoors.

So in this sentence, a native speaker may imagine:

• mud
• puddles
• wet dirt
• messy sidewalks or roads

That is why English translations often use muddy even though the Russian word is literally dirty.

The stress is:

• пóсле
• дождя́
• у́лице
• гря́зно

So the whole sentence is:

Пóсле дождя́ на у́лице гря́зно.

A rough pronunciation guide:

POS-lye dazh-DYA na OO-li-tse GRYAZ-na

A few notes:

• дождя́ has stress on the last syllable
• гря́зно has stress on the first syllable
• на улице is pronounced smoothly together in normal speech

Russian has no articles like a, an, or the. So дождь / дождя can mean:

• rain
• the rain
• a rain

The exact meaning depends on context.

In После дождя на улице грязно, English often uses the naturally: After the rain, it’s dirty outside. But Russian does not need a separate word for that.