Их отношения стали лучше.

Here их means their.

Russian их can mean either:

• them as an object: Я вижу их = I see them
• their as a possessive: их отношения = their relationship / their relations

In this sentence, их comes right before a noun (отношения), so it is naturally understood as their.

Russian very often uses отношения in the plural to talk about the state of a relationship between people.

So:

• Их отношения = their relationship / their relations

The singular отношение usually has a different meaning, more like attitude, relation, or approach depending on context. For example:

• моё отношение к этому = my attitude toward this

So in this sentence, plural отношения is the normal choice.

Because the subject отношения is plural.

In the past tense, Russian verbs agree with the subject in number, and in the singular also in gender.

• отношения стали = the relations became
• singular would be something like отношение стало

So стали matches plural отношения.

Because лучше here means better, and Russian normally uses the comparative form after стать when you mean become better.

So:

• стали лучше = became better / got better

Лучше is an indeclinable comparative form, so it does not change for gender, number, or case.

By contrast:

• лучшими means the best / better ones in instrumental plural, which does not fit here
• хорошие means good in plural, but стали хорошие would sound incomplete and unnatural in this sentence

It can mean both.

In natural English, this sentence could be translated as:

• Their relationship got better
• Their relations became better
• Things between them improved

Got better is often the most natural everyday translation.

It is in the nominative plural, because it is the subject of the sentence.

You can think of the structure as:

• Их отношения = the subject
• стали = the verb
• лучше = the complement meaning better

So the basic pattern is:

• Their relationship(s) became better

It is best thought of here as the comparative form meaning better.

Russian лучше can function in ways that overlap with both English better as an adjective and better as an adverb. In this sentence, it describes the state of the relationship, so in grammar explanations it is often treated as a predicative comparative.

The important practical point is:

• лучше = better
• it does not change form

Yes. That would mean:

• The relationship between them got better
• Relations between them improved

The difference is mainly in focus:

• Их отношения стали лучше = their relationship got better
• Отношения между ними стали лучше = emphasizes between them

Both are natural, but их отношения is shorter and very common.

Yes, but the given order is the most neutral:

• Их отношения стали лучше.

Russian word order is more flexible than English because cases and verb forms help show the roles of words.

Other orders are possible for emphasis, but they may sound less neutral. For example:

• Стали лучше их отношения. = more literary or emphatic
• Лучше стали их отношения. = marked emphasis on better

For a learner, the original order is the safest and most standard.

The main stress is:

• Их отношЕния стАли лУчше.

A rough pronunciation guide:

• их ≈ ikh
• отношения ≈ aht-na-SHE-ni-ya
• стали ≈ STA-li
• лучше ≈ LOO-chshe

A natural rhythm would be:

• Их отношЕния стали лУчше.

Yes.

Depending on context, их отношения can refer to:

• a romantic relationship
• friendship
• family relations
• professional relations
• general personal dynamics

So the sentence itself does not specify exactly what kind of relationship it is. The broader context tells you that.

Because Russian has no articles like a or the.

So их отношения can mean:

• their relationship
• their relations

The exact English wording depends on context. Russian simply does not mark definiteness with articles the way English does.