Друзьям иногда приходится извиняться друг перед другом после ссоры.

Because the construction кому-то приходится + infinitive uses the dative for the person who has to do something.

So here:

• друзьям = to the friends / for the friends
• приходится извиняться = have to apologize

A very literal breakdown is something like:

• Друзьям приходится извиняться = The friends have to apologize

Even though English uses a normal subject here, Russian treats this as an impersonal construction.

Here приходится means to have to, to be forced to, or to end up needing to.

It comes from приходиться, but in this meaning it does not mean physical motion like to come.

So:

• мне приходится работать = I have to work
• нам приходится ждать = we have to wait

In your sentence, иногда приходится gives the idea sometimes they have to.

Compared with должны, приходится often suggests necessity caused by circumstances rather than duty or moral obligation.

Because приходится here is part of an impersonal construction.

The word друзьям is in the dative case, so it is not the grammatical subject. Since there is no normal nominative subject, Russian uses the default singular form:

• мне приходится
• тебе приходится
• друзьям приходится

So the verb does not agree with друзьям.

In Russian, извиняться is the normal verb meaning to apologize.

The ending -ся is part of the verb itself here. Historically it is reflexive, but for a learner it is best to remember:

• извиняться = to apologize
• извинить = to forgive / excuse someone
• извинить меня = to excuse me / forgive me

So in this sentence, the friends are not excusing themselves in a literal English sense; they are simply apologizing.

This is an aspect question.

• извиняться = imperfective
• извиниться = perfective

The imperfective is natural here because the sentence is talking about something that happens sometimes, as a general or repeated situation:

• Друзьям иногда приходится извиняться...
  = Friends sometimes have to apologize...

If you used извиниться, it would sound more like a single completed act in a specific situation.

So извиняться fits the habitual meaning better.

Yes, and in Russian the verb often takes the pattern:

• извиняться перед кем? = to apologize to/before whom?

So Russian says literally something closer to apologize before someone.

That is why the sentence has:

• перед другом
• and in reciprocal form, друг перед другом

This is normal Russian verb government.

You can also have:

• извиняться за что? = to apologize for what?

For example:

• извиняться перед другом за грубость
  = to apologize to a friend for being rude

This is a reciprocal expression meaning before each other or, in natural English here, to each other.

Russian often builds each other expressions with друг...другом / друга / другу depending on the case required.

Examples:

• друг друга = each other
• друг другу = to each other
• друг с другом = with each other
• друг перед другом = before each other

Here the preposition перед requires the instrumental case, so the second word becomes:

• другом = instrumental singular of друг

So:

• друг перед другом = before each other

It sounds a bit literal in English, but with извиняться перед кем it gives the normal meaning apologize to each other.

Because после always takes the genitive case.

So:

• ссора = quarrel, argument
• после ссоры = after a quarrel / after an argument

This is:

• nominative: ссора
• genitive: ссоры

So the phrase is completely regular:

• после урока = after the lesson
• после работы = after work
• после ссоры = after a quarrel

They can both sometimes translate as have to, but the nuance is different.

• должны = obligation, duty, expectation, something one is supposed to do
• приходится = necessity caused by circumstances, something one ends up having to do

Compare:

• Друзья должны извиняться
  = Friends should / are supposed to apologize
• Друзьям приходится извиняться
  = Friends have to apologize because the situation makes it necessary

In your sentence, приходится sounds more natural because after a quarrel, apologizing is something that becomes necessary.

The word order is fairly natural and neutral, but Russian word order is flexible.

Друзьям иногда приходится извиняться друг перед другом после ссоры.

This order gives:

• Друзьям as the topic: we are talking about friends
• иногда modifies the whole situation: this happens sometimes
• после ссоры comes at the end as contextual information

You could move parts around for emphasis, for example:

• После ссоры друзьям иногда приходится извиняться друг перед другом.

That means basically the same thing, but it foregrounds after a quarrel.

Yes. The phrase друг перед другом clearly gives a mutual meaning.

So the idea is not just that one friend apologizes to the other, but that they apologize to each other.

If Russian wanted to talk about only one person apologizing to another, it would use a different structure, for example:

• Один друг извиняется перед другим.
  = One friend apologizes to the other.

But in your sentence, the reciprocity is explicit.