После работы я возвращаюсь домой пешком.

Because после always takes the genitive case in Russian.

The dictionary form is работа.
After после, it changes to the genitive singular: работы.

So:

• работа = work
• после работы = after work

This is something you usually just memorize with the preposition:

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

The -сь ending marks the verb as reflexive.

The verb here is возвращаться = to return / to come back.
Its first-person singular present form is я возвращаюсь = I return / I come back.

Compare:

• возвращать = to return something
• возвращаться = to return oneself, to come back

So:

• Я возвращаю книгу. = I return the book.
• Я возвращаюсь домой. = I return home.

In many cases, Russian uses a reflexive verb where English simply uses return.

Because Russian often uses the present tense to describe habitual or repeated actions, just like English can say:

• After work I go home on foot.
• After work I return home on foot.

So я возвращаюсь here does not have to mean I am returning right now. It can mean I usually return or I return as a routine.

Also, the verb is imperfective (возвращаться), which fits repeated, ongoing, or habitual actions.

If you wanted a single completed future action, you would more likely use a perfective form such as вернусь:

• После работы я вернусь домой пешком. = After work I’ll return home on foot.

Because домой is the special Russian word used for motion toward home.

Compare:

• дома = at home
• домой = homeward, to home

So:

• Я дома. = I am at home.
• Я иду домой. = I am going home.

You do not normally say к дому when you mean home in the everyday sense.
К дому literally means toward the house/building, not to my home in the usual idiomatic way.

So in this sentence, домой is exactly the natural choice because there is movement toward home.

Пешком means on foot.

For learners, the easiest way is to treat it as a fixed adverbial word meaning on foot. Historically, it is an instrumental-form expression, which is why it ends in -ом.

So:

• идти пешком = to go on foot
• возвращаться пешком = to return on foot

English speakers sometimes want to translate this as by foot, but natural English is on foot, and Russian uses пешком.

Examples:

• Я хожу в школу пешком. = I go to school on foot.
• Мы вернулись пешком. = We returned on foot.

Yes, but the meaning shifts slightly.

• после работы = after work
  This emphasizes time: when work is over.

• с работы = from work
  This emphasizes origin/source: the place you are coming from.

So:

• После работы я возвращаюсь домой пешком.
  = After work, I return home on foot.

• Я возвращаюсь домой с работы пешком.
  = I return home from work on foot.

Both can be natural, but they are not exactly the same:

• после работы answers when?
• с работы answers from where?

You can often omit it.

Russian verb endings usually show the subject clearly, and возвращаюсь already means I return.

So both are possible:

• После работы я возвращаюсь домой пешком.
• После работы возвращаюсь домой пешком.

The version with я can sound a bit more explicit, contrastive, or simply clearer in context. Russian often includes pronouns when the speaker wants emphasis or when it helps the flow.

No. Russian word order is fairly flexible.

The sentence as given is a neutral, natural order:

• После работы я возвращаюсь домой пешком.

But other orders are possible depending on emphasis:

• Я после работы возвращаюсь домой пешком.
  Slightly more focus on I.

• Домой я возвращаюсь пешком после работы.
  More marked; possible if home is being highlighted.

• Пешком я возвращаюсь домой после работы.
  Emphasizes on foot.

So word order in Russian is not only about grammar; it is also about information focus and style.

Because возвращаюсь emphasizes returning or coming back, not just moving.

Compare:

• иду домой = I’m going home / I go home
• возвращаюсь домой = I’m returning home / I return home

In this context, after work, возвращаюсь suggests that the speaker is going back home from somewhere else. It highlights the idea of coming back.

If you say После работы я иду домой пешком, that is also understandable and can be natural. But it focuses more on the action go home, while возвращаюсь more clearly expresses return home.

Yes. It is the first-person singular present form of возвращаться.

The pattern is:

• я возвращаюсь = I return
• ты возвращаешься = you return
• он/она возвращается = he/she returns
• мы возвращаемся = we return
• вы возвращаетесь = you return
• они возвращаются = they return

A common thing to notice is that the reflexive ending changes:

• after a vowel: usually -сь
• after a consonant: usually -ся

So here you get возвращаю-сь.

This is completely normal and something you will see with many reflexive verbs.