После прогулки я заметил, что подошва у одной кроссовки была мокрой и грязной.

Because после requires the genitive case.

• Dictionary form: прогулка
• Genitive singular: прогулки

So после прогулки means after the walk / after a walk.

This is a very common pattern:

• после урока — after the lesson
• после работы — after work
• после дождя — after the rain

Заметил is perfective, and that fits the idea of a single completed act of noticing.

Here the speaker is talking about one moment:

• first, the walk happened
• then, the speaker noticed something

So я заметил = I noticed.

If you used замечал, it would usually suggest repeated or habitual noticing, or an ongoing process:

• Я замечал это раньше — I had noticed this before / I used to notice this

Что introduces a subordinate clause, like English that in I noticed that...

So:

• я заметил — I noticed
• что подошва... была мокрой и грязной — that the sole... was wet and dirty

In Russian, что is normally used in this kind of sentence. English sometimes drops that, but Russian usually keeps что.

Both are possible, but they feel slightly different.

• подошва у одной кроссовки is a very natural Russian way to say the sole of one sneaker
• literally, у often means something like at / by / belonging to
• this structure is common in Russian when talking about parts of something:
  • у машины колесо — the car has a wheel / the wheel of the car
  • у стула ножка сломана — the chair has a broken leg

So подошва у одной кроссовки is a normal part-whole construction.

Подошва одной кроссовки is also correct, and sounds a bit more compact or formal.

Good question: кроссовки can look like a plural form, but here it is actually genitive singular.

The key is одной:

• одной кроссовки = of one sneaker
• одной is feminine singular genitive
• so кроссовки here must also be singular genitive

Compare:

• одна кроссовка — one sneaker
• нет одной кроссовки — there is no one sneaker / one sneaker is missing
• у одной кроссовки — of one sneaker

So in this sentence, it means one of the two sneakers, not plural sneakers.

Yes, in this sentence кроссовка is treated as a feminine noun.

That is why you get:

• одна кроссовка
• у одной кроссовки

The adjective/pronoun одной agrees with кроссовки in:

• gender: feminine
• number: singular
• case: genitive

This agreement is an important clue for understanding the grammar.

Because была agrees with подошва, not with кроссовки.

The subject of the clause is подошва:

• подошва — sole
• feminine singular

So the past-tense form must also be feminine singular:

• подошва была

Even though кроссовки appears nearby, it is not the grammatical subject here. It is part of the phrase у одной кроссовки.

After быть in the past tense, Russian often uses full adjectives in the instrumental case as part of the predicate:

• была мокрой и грязной

This is a normal pattern.

You can also hear:

• была мокрая и грязная

Both are possible. Very roughly:

• instrumental often sounds a bit more standard or traditional after быть
• nominative can sound more direct or conversational

So the sentence is perfectly normal as written, and мокрой и грязной agrees with подошва:

• feminine
• singular
• instrumental

Подошва means sole.

In this sentence it clearly means the sole of a shoe, because it is connected with кроссовка.

Russian подошва can also mean the sole of the foot in some contexts, but here the meaning is clearly the shoe sole.

Russian word order is flexible, so this order is mainly about focus and style, not basic meaning.

The sentence begins with:

• После прогулки — setting the time frame first

Then:

• я заметил — the main action
• что подошва у одной кроссовки... — what was noticed

Putting подошва early gives it some emphasis: the speaker is focusing on the sole as the thing that turned out to be wet and dirty.

A different order is also possible:

• После прогулки я заметил, что у одной кроссовки подошва была мокрой и грязной.

That still means the same thing, but the emphasis shifts slightly.

Yes. A few natural alternatives are:

• После прогулки я заметил, что подошва одной кроссовки была мокрой и грязной.
• После прогулки я заметил, что у одной кроссовки подошва была мокрой и грязной.

The original sentence is understandable and natural. These alternatives are just slightly different in structure:

• подошва одной кроссовки — a bit more compact
• подошва у одной кроссовки — very natural in everyday Russian, especially when talking about a part of an object