У моей сестры красивые кудри.

Russian usually expresses possession with the pattern у + genitive + possessed thing.

So:

• У моей сестры красивые кудри.
• literally: At my sister [there are] beautiful curls
• natural English: My sister has beautiful curls.

Russian does have the verb иметь (to have), but for ordinary possession it is often less natural and can sound formal or bookish. So у моей сестры... is the normal everyday way to say it.

Because the preposition у requires the genitive case.

The base form is:

• моя сестра = my sister

After у, both words change to genitive:

• моя → моей
• сестра → сестры

So:

• у моей сестры = at my sister / my sister has

Красивые кудри is in the nominative plural.

Why?

Because in this possession pattern, the person who possesses something goes after у in the genitive, but the thing possessed is usually in the nominative.

So here:

• у моей сестры = genitive phrase
• красивые кудри = nominative plural

And the adjective agrees with the noun:

• кудри = plural
• красивые = plural

Russian normally does not use a separate present-tense verb like English has in this structure.

Instead, possession is shown by:

• у
  • possessor in genitive
• the possessed thing

So У моей сестры красивые кудри literally feels like:

• At my sister [are] beautiful curls

Also, Russian usually omits the present-tense form of to be, so there is no extra word corresponding to is/are here either.

In Russian, кудри is very commonly used in the plural when talking about someone's curly hair or curls in general.

So:

• кудри = curls / curly hair

A singular curl can be expressed, but when describing a person's hair, the plural is much more natural. That is why the sentence uses кудри, not a singular form.

So the sentence means something like:

• My sister has beautiful curls
• or My sister has beautiful curly hair

Because красивые agrees with кудри, not with сестры.

This is a very common point of confusion:

• у моей сестры is genitive because of у
• красивые кудри is a separate part of the sentence, and кудри is nominative plural

So the adjective must match кудри:

• nominative plural кудри
• nominative plural красивые

If it were красивых, that would be genitive/accusative plural, which is not what is needed here.

Yes, that is grammatical.

Adding есть can make the idea of possession a little more explicit:

• У моей сестры есть красивые кудри.

But in many situations, Russian simply leaves есть out, especially when the sentence is descriptive and natural in context:

• У моей сестры красивые кудри.

Both are possible, but the version without есть is very normal and smooth.

Yes. Russian word order is flexible, and changing it changes the emphasis more than the basic meaning.

The neutral order here is:

• У моей сестры красивые кудри.

This presents the possessor first and then says what she has.

You might also hear other orders in context, but they would sound more marked or literary, for example if the speaker wants to emphasize beautiful curls or contrast my sister with someone else.

For a learner, У моей сестры красивые кудри is the best standard pattern to remember.

It is understandable, but it is not the most natural everyday Russian.

Grammatically, иметь means to have, so the sentence works. But for ordinary possession, Russian usually prefers the у + genitive structure:

• more natural: У моей сестры красивые кудри.
• less natural in everyday speech: Моя сестра имеет красивые кудри.

So if you want to sound natural, use у моей сестры rather than имеет here.