У дочки новый фломастер, и она рисует им солнце в альбоме.

Because у requires the genitive case when it is used to show possession.

• дочь = daughter (dictionary form, nominative)
• дочки = of the daughter / by the daughter (genitive singular here)

So У дочки новый фломастер literally means something like By the daughter, there is a new marker, which is how Russian commonly expresses The daughter has a new marker.

Also, дочка is a more everyday, affectionate form of дочь.

Russian usually does not use a verb equivalent to English have in this kind of sentence. Instead, it often uses this pattern:

• у + person in the genitive + thing
• literally: at/by someone there is something

So:

• У дочки новый фломастер = The daughter has a new marker

This is one of the most important possession patterns in Russian.

In the present tense, есть is often omitted in this possession structure.

So both of these can exist:

• У дочки есть новый фломастер
• У дочки новый фломастер

Both mean roughly The daughter has a new marker, but omitting есть is very common and natural.

Because новый describes фломастер, not дочка.

• фломастер is masculine
• so the adjective must also be masculine: новый

If the noun were feminine, then the adjective would be feminine too.

So:

• новый фломастер = a new marker
• not новая, because that would have to agree with a feminine noun

Фломастер usually means felt-tip pen or marker.

Depending on context, English might translate it as:

• marker
• felt-tip pen
• sometimes colored marker

In this sentence, it is the tool she is using to draw.

Because Russian uses the instrumental case for the tool used to do something.

Here the idea is:

• рисовать чем? = to draw with what?
• фломастером = with a marker

When that tool is replaced by a pronoun, the pronoun also goes into the instrumental:

• он → им

So:

• она рисует им = she is drawing with it

Since фломастер is masculine, the pronoun is based on он.

Because дочка is the most natural feminine noun available as the antecedent, and the meaning fits perfectly:

• дочка = feminine
• она = she

So она рисует naturally means the daughter is drawing.

Russian often uses pronouns this way once the person has already been introduced.

Солнце is a direct object, so it is in the accusative case. But for many inanimate nouns, the accusative looks exactly the same as the nominative.

Here:

• nominative: солнце
• accusative: солнце

So the case is accusative by function, even though the form does not change.

It means in the album/sketchbook/notebook.

The preposition в can mean in, and because this is a location, Russian uses the prepositional case:

• альбом → в альбоме

In context, альбом often means a drawing pad, sketchbook, or album-like notebook, not necessarily only a photo album.

It can mean both, depending on context.

Russian present tense often covers both:

• she draws
• she is drawing

Here, because the sentence describes what she is doing with the new marker, English would often naturally say:

• she is drawing

But grammatically Russian just uses the present tense form рисует.

Russian has no articles. It does not have direct equivalents of English a/an and the.

So:

• дочки could mean the daughter, a daughter, or my/our daughter, depending on context
• новый фломастер could mean a new marker or the new marker

The exact meaning comes from context, not from an article.

Yes. Russian word order is fairly flexible, and changing it often changes emphasis rather than basic meaning.

This sentence is neutral and natural:

• У дочки новый фломастер, и она рисует им солнце в альбоме.

But other orders are possible, for example to emphasize the tool, the picture, or the place.

So Russian word order is not random, but it is more flexible than English.