Моя сестра замужем.

In Russian, the present-tense form of to be is usually omitted.

So:

• Моя сестра замужем. = My sister is married.

Russian does have быть for to be, but in the present tense you normally do not say it here.

Compare:

• Моя сестра замужем. = My sister is married.
• Моя сестра была замужем. = My sister was married.
• Моя сестра будет замужем. = My sister will be married.

So the missing is is completely normal.

Because моя has to agree with сестра, and сестра is a feminine noun.

Russian possessives change form to match the gender and number of the noun:

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

Since сестра is feminine singular, the correct form is моя.

It is in the nominative case because it is the subject of the sentence.

The sentence is about the sister, so сестра is the thing being described:

• Моя сестра = the subject
• замужем = what is being said about her

So сестра stays in its dictionary form here.

Замужем is the normal Russian word used to say that a woman is married.

It is used predicatively, meaning it describes someone’s state:

• Она замужем. = She is married.
• Моя сестра замужем. = My sister is married.

A useful point for learners: Russian uses different words for men and women here.

• Он женат. = He is married.
• Она замужем. = She is married.

So for a woman, замужем is the standard word.

No, not in standard Russian.

For marital status:

• for a man: женат
• for a woman: замужем

So these are correct:

• Мой брат женат.
• Моя сестра замужем.

A native English speaker may expect one word meaning married for everyone, but Russian splits this by gender in this common expression.

With замужем, you normally use за + instrumental.

Examples:

• Моя сестра замужем за врачом. = My sister is married to a doctor.
• Она замужем за Иваном. = She is married to Ivan.

So the pattern is:

• замужем за кем? = married to whom?

The noun after за goes in the instrumental case.

Just put не before замужем:

• Моя сестра не замужем. = My sister is not married.

This is the normal way to negate it.

Yes. Russian word order is more flexible than English word order.

The neutral, straightforward version is:

• Моя сестра замужем.

But other orders are possible for emphasis or style, for example:

• Сестра моя замужем.
• Замужем моя сестра.

These can sound more marked, poetic, or context-dependent. For most learners, Моя сестра замужем is the best basic order to use.

A helpful stress pattern is:

• моя́ сестра́ за́мужем

Approximate pronunciation:

• ma-YA syes-TRA ZA-mu-zhem

A few notes:

• я in моя́ is stressed.
• а in сестра́ is stressed.
• за́мужем is stressed on the first syllable.
• ж sounds like the s in measure.

Because Russian has no articles like a/an/the.

So Russian does not mark that difference directly with separate words.

• сестра can mean sister, a sister, or the sister, depending on context.
• моя сестра naturally means my sister.

English requires an article in many places, but Russian does not.

Yes, if the context already makes it clear whose sister you mean.

• Сестра замужем. = My/The sister is married, depending on context.
• Моя сестра замужем. is clearer when you are introducing the idea for the first time.

So моя is not grammatically required for the sentence structure, but it is required if you specifically want to say my sister.