دیروز داشتم پیاده به خانه میآمدم که همسایه ام را در خیابان دیدم.

Yes, together they create the idea of an action that was in progress in the past.

• داشتم = I was having / an auxiliary used for the past progressive
• می‌آمدم = I was coming / used to come

In this sentence, داشتم ... می‌آمدم means I was in the middle of coming or more naturally I was walking home.

So:

• می‌آمدم alone can already mean I was coming or I used to come, depending on context.
• داشتم می‌آمدم makes it clearly progressive, like English I was coming / I was on my way.

This is a very common pattern in Persian:

• داشتم می‌رفتم = I was going
• داشتم می‌خواندم = I was reading
• داشتم کار می‌کردم = I was working

In your sentence, this ongoing action sets the background for the shorter event دیدم = I saw.

Here پیاده means on foot or walking.

So:

• پیاده به خانه می‌آمدم = I was coming home on foot
• more naturally: I was walking home

It is not functioning as a full verb here. It is more like an adverbial expression describing how the person was going home.

Some useful comparisons:

• پیاده رفتم = I went on foot
• پیاده آمد = He/She came walking
• با ماشین رفتم = I went by car

So پیاده answers the question How were you coming home? → On foot.

Persian word order is flexible, but this order is natural.

• پیاده = on foot
• به خانه = to home / homeward
• می‌آمدم = I was coming

So literally it is something like:

• I was on foot to home coming

A more natural English translation is:

• I was walking home

In Persian, adverbial elements like manner and destination often come before the verb. So پیاده به خانه می‌آمدم sounds normal.

Other possible orders may also be understood, such as:

• به خانه پیاده می‌آمدم
• پیاده می‌آمدم به خانه

But the original version is quite natural and smooth.

به means to, and خانه means home/house.

So literally:

• به خانه می‌آمدم = I was coming to home

But in English we usually just say:

• I was coming home

Persian often uses به before destinations:

• به مدرسه رفتم = I went to school
• به خانه برگشتم = I returned home
• به تهران آمدم = I came to Tehran

So even if English sometimes drops to, Persian normally keeps به.

That is a very common question. که often does mean that, but in this kind of sentence it can also mean something like when, and then, or at which point.

Here:

• دیروز داشتم پیاده به خانه می‌آمدم که ... دیدم means:
• Yesterday I was walking home when ... I saw ...

So که links the ongoing background action to the new event that happened during it.

You can think of it here as introducing an interrupting event:

• I was doing X, when Y happened.

This use is very natural in Persian.

Because دیدم is a completed event: I saw.

The sentence contrasts:

• an ongoing background action: داشتم ... می‌آمدم = I was coming / walking
• a single event that happened during it: دیدم = I saw

This is similar to English:

• I was walking home when I saw my neighbor.

If you said می‌دیدم, it would usually suggest a repeated, habitual, or ongoing kind of seeing, which would not fit as well here.

So the aspect contrast is important:

• داشتم پیاده به خانه می‌آمدم = background, in progress
• همسایه‌ام را در خیابان دیدم = completed event

را marks a specific direct object.

So:

• همسایه‌ام را دیدم = I saw my neighbor

Because my neighbor is definite and specific, Persian uses را.

Compare:

• کتاب را خواندم = I read the book
• دوستم را دیدم = I saw my friend
• آن مرد را می‌شناسم = I know that man

Without را, the sentence would sound incomplete or less standard in this context.

So را is one of the clearest signs that همسایه‌ام is the object of دیدم.

The base noun is:

• همسایه = neighbor

Then Persian adds the enclitic -ام to mean my:

• همسایه‌ام = my neighbor

This is extremely common in Persian:

• دوست = friend → دوستم = my friend
• خانه = house/home → خانه‌ام = my home/house
• اسم = name → اسمم = my name

You may see it written with a half-space:

• همسایه‌ام

Sometimes learners also see looser spacing in informal typing:

• همسایه ام

But همسایه‌ام is the standard written form.

Because Persian verbs already show the subject through their endings.

For example:

• داشتم = I was having
• می‌آمدم = I was coming
• دیدم = I saw

The ending -م shows I.

So Persian often omits the subject pronoun when it is clear from the verb. This is very normal.

You could say من for emphasis:

• من دیروز داشتم ... but it is not necessary.

This is one reason Persian sentences can seem compact compared with English.

در خیابان means in the street or on the street.

• در = in
• خیابان = street

So:

• همسایه‌ام را در خیابان دیدم = I saw my neighbor in the street

Yes, in everyday speech Persian sometimes drops در in some expressions, depending on style and context, but در خیابان is perfectly standard and clear.

English says in the street or on the street, but Persian uses در.

دیروز means yesterday, and Persian often places time expressions near the beginning of the sentence.

So:

• دیروز داشتم ... = Yesterday I was ...

This is very natural because it sets the time frame first.

Persian allows some flexibility, though. You could also hear:

• داشتم دیروز پیاده به خانه می‌آمدم ... but the original version is much more natural.

Starting with دیروز helps frame the whole event immediately.

Yes, it comes from آمدن = to come.

The past stem is:

• آمد-

So:

• آمدم = I came
• می‌آمدم = I was coming / I used to come

The structure is:

• می‌-
  • past stem + personal ending

So:

• می‌آمدم = می‌
  • آمد
    • م

It may look a little unusual because آمدن is a very common verb with forms learners need to get used to.

Also, in careful writing, it is usually written with the half-space:

• می‌آمدم

It is basically standard written/spoken Persian, fairly neutral.

A more colloquial spoken version might be something like:

• دیروز داشتم پیاده می‌اومدم خونه که همسایه‌م رو تو خیابون دیدم.

Compared with the original:

• می‌آمدم → colloquial می‌اومدم
• خانه → colloquial خونه
• را → colloquial رو
• در خیابان → often تو خیابون

So the sentence you have is a good standard-learning version, while everyday speech may sound more relaxed.

A very literal breakdown would be:

• دیروز = yesterday
• داشتم = I was
• پیاده = on foot
• به خانه = to home
• می‌آمدم = coming
• که = when
• همسایه‌ام را = my neighbor
• در خیابان = in the street
• دیدم = I saw

Very literal:

• Yesterday I was coming home on foot when I saw my neighbor in the street.

More natural English:

• Yesterday I was walking home when I saw my neighbor in the street.

The sentence shows a very common storytelling pattern in Persian:

ongoing background action + که + short completed event

In this sentence:

• داشتم پیاده به خانه می‌آمدم = ongoing background
• که همسایه‌ام را در خیابان دیدم = interrupting event

This is the Persian equivalent of English:

• I was doing X when Y happened.

It is a useful pattern to learn because it appears constantly in narratives:

• داشتم غذا می‌خوردم که تلفن زنگ زد.
  • I was eating when the phone rang.
• داشتم درس می‌خواندم که دوستم آمد.
  • I was studying when my friend came.

So your sentence is a very good example of how Persian handles background action plus a new event.