Моя сестра умеет утешать людей и искренне сочувствовать им, когда им тяжело.

Because сестра is a feminine singular noun, and моя is the feminine singular form of мой.

Russian possessive words like мой / моя / моё / мои must agree with the noun in:

• gender
• number
• case

So:

• мой брат = my brother
• моя сестра = my sister
• моё письмо = my letter
• мои друзья = my friends

Here, сестра is feminine singular nominative, so моя is required.

Умеет comes from уметь, which means to know how to, to be able to because of skill, or to have the ability.

So моя сестра умеет утешать людей means something like:

• My sister knows how to comfort people
• My sister is good at comforting people

This is different from может from мочь, which usually means:

• can
• is able to
• is possible for someone in a particular situation

Compare:

• Она умеет плавать. = She knows how to swim.
• Она может плавать сегодня. = She can swim today / She is able to swim today.

In this sentence, умеет is used because it is talking about a general personal ability, not a one-time possibility.

Because after уметь, Russian normally uses the infinitive.

Pattern:

• уметь + infinitive

So:

• умеет утешать
• умеет сочувствовать

This is similar to English know how to do something or be able to do something.

Examples:

• Он умеет готовить. = He knows how to cook.
• Она умеет слушать. = She knows how to listen.

Here the sentence has two infinitives joined by и:

• утешать людей
• искренне сочувствовать им

Both depend on умеет.

They are imperfective because the sentence describes a general ability or habitual skill, not one completed action.

Russian often uses the imperfective infinitive after уметь when talking about what someone generally knows how to do.

So:

• уметь утешать = to know how to comfort people
• уметь сочувствовать = to know how to sympathize

If you used perfective forms, it would sound much less natural here, because perfective usually points to a single completed result, while this sentence is about a repeated or ongoing personal quality.

Because людей is the accusative plural form of люди, and with animate plural nouns, the accusative looks like the genitive.

The verb утешать takes a direct object, so people must be in the accusative:

• nominative: люди
• accusative plural: людей

This happens with animate nouns in Russian:

• Я вижу людей. = I see people.
• Она любит детей. = She loves children.

But with inanimate plural nouns, accusative usually looks like nominative:

• Я вижу столы. = I see tables.

So утешать людей is correct because people are animate and are the direct object.

Because сочувствовать requires the dative case.

So:

• сочувствовать кому? = to sympathize with whom?
• им = to them

This is a very common thing in Russian: many verbs do not take a direct object the way their English equivalents do. Instead, they govern a specific case.

Examples:

• помогать кому = to help someone
• верить кому = to believe someone
• звонить кому = to call someone
• сочувствовать кому = to sympathize with someone

So:

• сочувствовать им = to sympathize with them

This им is there for a different grammatical reason.

In когда им тяжело, the word тяжело means it is hard / difficult, and Russian uses the dative to show the person experiencing the feeling or situation.

So:

• им тяжело = it is hard for them / they are having a hard time

This is an impersonal structure:

• мне трудно = it is difficult for me
• ему плохо = he feels bad / it is bad for him
• нам холодно = we are cold / it is cold to us
• им тяжело = they are struggling / it is hard for them

So the two им are not doing the same job:

• сочувствовать им: dative after the verb сочувствовать
• им тяжело: dative experiencer with тяжело

Yes. The verb умеет applies to both infinitives.

The structure is:

• Моя сестра умеет
  • утешать людей
  • и искренне сочувствовать им

This is very normal in Russian. When two infinitives are connected by и, the first governing verb can control both of them.

In English, we would also do this:

• My sister knows how to comfort people and sincerely sympathize with them

You do not need to repeat умеет.

Искренне means sincerely or genuinely.

It modifies сочувствовать, so:

• искренне сочувствовать = to sympathize sincerely / genuinely

Its position before the verb is natural and common. Russian word order is flexible, but this placement sounds smooth and neutral.

Possible alternatives can exist, but they may change emphasis slightly:

• искренне сочувствовать им
• сочувствовать им искренне

The version in your sentence is the most natural for ordinary neutral speech.

Because in the present tense, Russian usually does not use the verb to be.

So where English says:

• It is hard for them

Russian simply says:

• им тяжело

This is a standard feature of Russian:

• Он врач. = He is a doctor.
• Я дома. = I am at home.
• Нам интересно. = It is interesting for us / We are interested.
• Им тяжело. = It is hard for them.

So nothing is missing; this is just how Russian works in the present tense.

Here тяжело does not mean physically heavy. It means:

• hard
• difficult
• painful
• emotionally tough

In когда им тяжело, it usually means when they are going through a hard time.

Russian тяжело can be used:

• physically: Мне тяжело идти. = It is hard for me to walk.
• emotionally: Ей тяжело после этого разговора. = She is having a hard time after that conversation.
• generally: Сейчас им тяжело. = Things are hard for them now.

So in this sentence, it most likely refers to people being in emotional difficulty, which fits well with comforting and sympathizing.

Because когда им тяжело is a subordinate clause.

The main clause is:

• Моя сестра умеет утешать людей и искренне сочувствовать им

The subordinate clause is:

• когда им тяжело

Russian normally separates subordinate clauses with a comma, especially when introduced by words like:

• когда = when
• если = if
• потому что = because
• что = that

So the comma here is required.

Yes. Им refers to the same people as людей.

So the sentence means:

• my sister knows how to comfort people
• and sincerely sympathize with them
• when they are having a hard time

Russian often repeats a pronoun like this even when the referent is obvious, because each verb or structure needs its own grammatical form:

• утешать людей → object of утешать
• сочувствовать им → dative after сочувствовать
• когда им тяжело → dative experiencer with тяжело

So the repetition is normal and grammatically necessary.