У сестры новая кофта с длинными рукавами, и она часто носит её дома.

Russian very often expresses possession with the pattern у + genitive.

So У сестры новая кофта literally feels like At the sister, there is a new sweater/top, but the natural English meaning is The sister has a new sweater/top.

This is one of the main ways Russian says have.

In the present tense, Russian often leaves out the verb to be.

So instead of saying something like At the sister there is a new sweater, Russian simply says:

У сестры новая кофта

You can also say У сестры есть новая кофта, where есть adds a clearer sense of there is / she has, but it is not always necessary.

Because the preposition у requires the genitive case in this possession pattern.

• сестра = nominative
• сестры = genitive singular

So:

• сестра = sister
• у сестры = of the sister / at the sister / the sister has

This is completely normal after у.

Кофта is the thing being talked about, so it appears in the nominative case.

• кофта is a feminine singular noun
• новая is the adjective new, and it must agree with кофта

So the forms match:

• feminine
• singular
• nominative

That is why you get новая кофта.

Кофта is a somewhat broad everyday word for a woman’s upper garment. Depending on context, it can mean something like:

• a sweater
• a knit top
• a cardigan-like top
• sometimes a blouse-like top

It does not usually mean a coat or jacket. In this sentence, top or sweater is a reasonable way to understand it.

Because с in the meaning with normally takes the instrumental case.

So:

• рукава = sleeves
• рукавами = with sleeves / by means of sleeves (instrumental plural)

The adjective must agree too:

• длинные = long
• длинными = long, instrumental plural

So с длинными рукавами means with long sleeves.

The plural is natural because clothes normally have two sleeves.

Grammatically, yes — both сестра and кофта are feminine singular nouns, so она could in theory refer to either one.

But the rest of the sentence makes the meaning clear:

• она часто носит её = she often wears it

A person wears clothing, so она is understood as the sister, and её refers to the sweater/top.

If someone wanted to avoid even that small ambiguity, they could repeat the noun:

...и сестра часто носит её дома.

Because её is the feminine singular form meaning her or it when used as an object.

Here it refers to кофта, which is a feminine noun, so Russian uses the feminine pronoun.

In this sentence:

• она = she
• её = it

More specifically, её is the accusative form here, because it is the direct object of носит.

Because носить means to wear, while надевать means to put on.

So:

• она носит её дома = she wears it at home
• она надевает её дома = she puts it on at home

Since the sentence describes a habitual action with часто (often), носит is the natural choice.

Дома is a very common adverb meaning at home.

So:

• дома = at home
• в доме = in the house/building

В доме focuses more on being physically inside a building.
Дома focuses on the idea of home as the place where someone lives.

That is why она часто носит её дома means she often wears it at home.

No. Russian word order is fairly flexible, and changes in word order often change emphasis, not the basic meaning.

This sentence has a neutral, natural order:

У сестры новая кофта с длинными рукавами, и она часто носит её дома.

But other orders are possible, for example:

• Дома она часто носит её = emphasizes at home
• Её она часто носит дома = emphasizes it
• Новая кофта у сестры = emphasizes the new sweater

So the original order is just the most straightforward way to present the information.