Learn Hardcore Russian: Её сестра вышла замуж позже и научилась быстро мириться после любой ссоры. - Her sister got married later and learned to make peace quickly after any quarrel.

жениться – used for a man:
- Он женился на Марии. – He married Maria.
выйти замуж – used for a woman:
- Она вышла замуж за Ивана. – She married Ivan.

So for сестра (female), you must say вышла замуж, not жениться or пожениться.
Пожениться is normally used for a couple together:

Они поженились. – They got married (as a couple).

Here we are talking about just the sister, so вышла замуж is the natural choice.

What exactly is замуж? Is it a preposition, a noun, or something else?

Замуж is an adverb-like form that historically comes from за мужа (“behind/for a husband”).

Today it’s a fixed word used almost only in the phrase выйти замуж (за кого?) – to get married (for a woman).
It doesn’t change its form and doesn’t behave like a normal noun anymore.
You cannot say *она замуж Ивана; you must keep the whole construction выйти замуж за Ивана.

So you can think of замуж as a fixed adverbial element that only really works with выйти / выходить in this meaning.

Why is the verb вышла (perfective) used here? Could it be выходила замуж instead?

Вышла is the past tense of perfective выйти, which presents the action as a single, completed event. Getting married is normally seen as a one-time completed act, so perfective is standard:

Её сестра вышла замуж позже. – She (at some point) got married later.

Выходила замуж (imperfective) is possible but has a different nuance, often suggesting:

repeated or attempted actions,
background description rather than a single finished event.

For example:

Она уже два раза выходила замуж. – She has already been married twice.

In your sentence, we just state the fact that this happened once, so вышла замуж is the natural, neutral choice.

Why is it научилась and not училась? What’s the difference between учиться and научиться?

The difference is aspect and result:

учиться (imperfective) – to study, to be learning; process, no guarantee of result.
- Она училась мириться. – She was learning (trying) to make peace.
научиться (perfective) – to learn something successfully; to acquire a skill.
- Она научилась мириться. – She learned / has learned how to make peace (and now knows how).

In научилась быстро мириться, the idea is that she acquired the ability to reconcile quickly. Using училась would focus on the process of learning, not on the fact that she actually mastered it.

Why is the infinitive мириться used after научилась?

In Russian, научиться is typically followed by an infinitive that expresses what you learned to do:

научиться + infinitive
- научилась быстро мириться – (she) learned to make peace quickly
- научился говорить по-русски – (he) learned to speak Russian
- научились водить машину – (they) learned to drive a car

So мириться is in its infinitive form because it’s the action she learned to perform.

What is the nuance of мириться here? Why not помириться?

Both verbs relate to “making peace,” but their aspect and usual usage differ:

мириться – imperfective, “to make peace / to reconcile” in general; can refer to the process or a habitual action.
- Она научилась быстро мириться. – She learned how to (habitually) make peace quickly.
помириться – perfective, “to make peace once,” focusing on a single completed reconciliation.
- Они помирились. – They made up (this particular time).

Since научиться expresses acquiring a general skill or habit, the infinitive that follows is usually imperfective. So научилась мириться (general ability), not *научилась помириться (which would sound wrong).

Why is it после любой ссоры and not после любая ссора? Which case is used?

The preposition после always takes the genitive case:

после чего? – after what?

So:

любая ссора – nominative (dictionary form)
любой ссоры – genitive (feminine singular)

Therefore:

после любой ссоры – after any quarrel/argument

Both the adjective (любой) and the noun (ссоры) must be in the genitive singular to agree with each other and with после.

What is the difference between любой and каждой in после любой ссоры? Could we say после каждой ссоры?

Both are grammatically possible, but they have slightly different nuances:

любой – any (doesn’t matter which one)
- после любой ссоры – after any quarrel; no matter what kind of quarrel it is.
каждый – each/every (emphasizes all individual instances)
- после каждой ссоры – after every quarrel; each time they argue, without exception.

In many real contexts they overlap, but:

любой sounds more like “it doesn’t matter what kind of fight, she can make peace quickly.”
каждой emphasizes the regularity: “every time they fight, she makes peace quickly.”

Why is the adverb быстро placed between научилась and мириться? Could we put it somewhere else?

Russian word order is relatively flexible, and adverbs like быстро can move:

Possible variants:

научилась быстро мириться – neutral; “learned to make peace quickly.”
быстро научилась мириться – “quickly learned to make peace” (emphasizes how fast she learned).
научилась мириться быстро – also possible, but a bit less neutral; it may sound like a slight afterthought.

The version in your sentence, научилась быстро мириться, emphasizes the manner of reconciliation (she reconciles quickly), not the speed of learning.

Why do the verbs end in -ла: вышла, научилась? What do these endings show?

In Russian, past tense verbs agree in gender and number:

masculine singular: -л (e.g. вышел, научился)
feminine singular: -ла (e.g. вышла, научилась)
neuter singular: -ло
plural (any gender mix): -ли

Since the subject is сестра (feminine singular), the verbs are:

вышла (замуж) – she (fem.) got married
научилась (мириться) – she (fem.) learned

So -ла marks feminine singular in the past tense.

Could we omit Её and just say Сестра вышла замуж позже? Would the meaning change?

Yes, you can say:

Сестра вышла замуж позже и научилась быстро мириться после любой ссоры.

The basic meaning remains “(The) sister got married later and learned to make peace quickly after any quarrel.”

The difference:

Её сестра – “Her sister” (explicitly tied to some earlier-mentioned woman).
Сестра – “(The) sister” in a more general sense; context would have to make clear whose sister it is, or it could sound like you’re speaking from your own point of view (“my sister”) in some contexts.

In isolation, Её сестра is clearer: it unambiguously means “her sister.”