После ссоры мы извиняемся друг перед другом и быстро миримся.

Because после requires the genitive case.

• ссора = nominative singular
• ссоры = genitive singular

So:

• после ссоры = after an argument / after the quarrel

This is a very common pattern in Russian:

• после урока = after the lesson
• после работы = after work
• после встречи = after the meeting

Also, Russian has no articles, so после ссоры can mean after an argument or after the argument, depending on context.

Because извиняться means to apologize, while извинять / извинить without -ся means to excuse / forgive someone.

So there is an important difference:

• Мы извиняемся. = We apologize.
• Мы извиняем его. = We excuse him / forgive him.

The -ся here is not really translated as ourselves in English. It is just part of the verb извиняться, which is the normal verb for to apologize.

Because Russian often expresses each other with the pattern друг + required case/preposition.

The exact form depends on what the verb needs.

Here the verb is:

• извиняться перед кем-то = to apologize to someone

Literally, Russian uses перед here, so to say to each other you match that pattern:

• друг перед другом = to each other / before each other

Other examples:

• любить друг друга = to love each other
• помогать друг другу = to help each other
• говорить друг с другом = to talk with each other

So друг перед другом is not random; it is built from the grammar of извиняться перед кем-то.

It is the instrumental singular.

That is because перед normally takes the instrumental case when it means before / in front of in this kind of expression.

So:

• перед другом = before a friend
• друг перед другом = before each other

In this reciprocal construction, the first друг stays unchanged, while the second part shows the case needed by the verb or preposition.

In Russian, the present tense is often used for a habitual, general, or typical action.

So this sentence sounds like:

• After a quarrel, we apologize to each other and quickly make up.

It describes what usually happens, not necessarily one specific past event.

If you wanted to talk about one completed event in the past, you would usually say something like:

• После ссоры мы извинились друг перед другом и быстро помирились.

This is about aspect.

• мириться = imperfective
• помириться = perfective

In the sentence, the verbs are in the present tense and describe a usual repeated action. For that, Russian normally uses the imperfective:

• мы миримся = we make up / we reconcile

The perfective помириться does not have a true present meaning. Forms like помиримся usually mean future:

• мы помиримся = we will make up

So миримся is the right choice for a general present-time statement.

Yes. In this sentence, that meaning is understood automatically.

Because the subject is мы and the verb мириться means to reconcile / make peace, the idea with each other is already implied.

Russian could also say:

• После ссоры мы извиняемся друг перед другом и быстро миримся друг с другом.

But that sounds more explicit and slightly heavier. In many contexts, just миримся is natural and enough.

Here миримся means we make up, we reconcile, or we become friends again after a quarrel.

The verb is мириться.

Depending on context, it can mean:

• to make peace / reconcile
• sometimes to put up with / accept something, especially with с:
  • мириться с трудностями = to put up with difficulties

But in this sentence, because of После ссоры and the general context, the meaning is clearly make up after an argument.

The main stresses are:

• по́сле
• ссо́ры
• извиня́емся
• друг
• перед
• дру́гом
• бы́стро
• мири́мся

So the sentence is pronounced roughly like:

ПО-сле ССО-ры мы из-ви-НЯ-ем-ся друг пе-РЕД ДРУ-гом и БЫ-стро ми-РИМ-ся.

Two especially useful ones to remember are:

• извиня́ться
• мири́ться

These stresses are very common and worth learning early.