İki komşu uzun süre dargın kaldı, sonra pazarda karşılaşıp güldü.

In Turkish, after a number, the noun is usually singular. So iki komşu is the normal way to say two neighbors.

• bir komşu = one neighbor
• iki komşu = two neighbors
• üç komşu = three neighbors

Using komşular after iki would sound unnatural in standard Turkish.

This is very common in Turkish. When the subject includes a number, the verb is often left in the singular:

• İki komşu geldi.
• Üç öğrenci bekliyor.

So İki komşu ... kaldı ... güldü is completely normal.

You may also hear plural verbs in some contexts, especially if the speaker wants to emphasize the people as individuals:

• İki komşu uzun süre dargın kaldılar.

That is possible, but the singular is very natural and common.

Dargın means something like offended, hurt, cross, or not on good terms. It often suggests that two people have had some kind of disagreement and are no longer speaking warmly to each other.

So it is not exactly the same as simple anger. It often has a relationship nuance: the people are emotionally distant or upset with each other.

Here dargın is an adjective, and kalmak means to remain / stay. Together, dargın kalmak means to remain offended or to stay on bad terms.

So the structure is:

• dargın = offended, cross
• kalmak = to remain, stay
• dargın kaldı = stayed offended / remained on bad terms

This is a very common Turkish pattern: adjective + kalmak.

For example:

• sessiz kaldı = stayed silent
• açık kaldı = stayed open
• yalnız kaldı = remained alone

Uzun süre means for a long time.

Literally:

• uzun = long
• süre = duration, period of time

It functions as a time expression telling you how long they remained dargın.

So:

• uzun süre dargın kaldı = stayed on bad terms for a long time

Karşılaşıp comes from karşılaşmak, which means to meet / run into each other / encounter each other.

The ending -ıp / -ip / -up / -üp is a converb ending. It links one action to another and often means something like:

• having done X, ...
• and then ...
• by doing X, ...

So:

• karşılaşıp güldü = they met and laughed
• more literally: meeting each other, they laughed

Because of vowel harmony, the ending here is -ıp, giving karşılaşıp.

Yes. In this kind of structure, the usual expectation is that the same subject performs both actions.

So in İki komşu ... karşılaşıp güldü, the two neighbors are the ones who:

1. karşılaştı = met each other
2. güldü = laughed / smiled

This is one reason the -ıp structure is so useful in Turkish: it neatly connects actions done by the same subject.

Because -da / -de is the locative ending, meaning in / at / on depending on context.

• pazar = market
• pazarda = at the market / in the market

So pazarda karşılaşıp means meeting each other at the market.

The form is -da here because of vowel harmony and consonant choice.

It can mean either, depending on context.

The verb gülmek commonly means to laugh, but in some contexts it can also be understood as to smile.

In this sentence, both are possible in translation depending on the tone:

• laughed if the idea is that they happily laughed after meeting
• smiled if the idea is softer, more like a warm reaction after a long disagreement

Turkish often leaves that nuance to context.

Yes. Karşılaşmak already has a reciprocal sense: it means to encounter one another / meet each other. So English translations like these are all reasonable:

• they met at the market
• they ran into each other at the market
• they encountered each other at the market

English often needs extra words to express what Turkish packs into one verb.

Turkish word order is flexible. The sentence as given is very natural and neutral:

• İki komşu uzun süre dargın kaldı, sonra pazarda karşılaşıp güldü.

But other orders are possible if you want to emphasize a different part:

• İki komşu pazarda sonra karşılaşıp güldü would sound less natural because sonra is awkwardly placed.
• Sonra iki komşu pazarda karşılaşıp güldü puts more focus on then.
• Pazarda iki komşu karşılaşıp güldü puts more focus on at the market.

So the order is not completely fixed, but the original version is a very natural default.