Дочке нравятся спагетти с сыром, а мне — с овощным соусом.

Because the verb нравиться works differently from English to like.

In Russian, the person who experiences the liking is put in the dative case:

• дочке = to the daughter
• мне = to me

Grammatically, the thing being liked is the subject of the sentence, not the person.

So the structure is closer to:

• Спагетти нравятся дочке = Spaghetti are pleasing to the daughter

That is why you see дочке and мне, not дочка and я.

Because нравиться agrees with the thing that is liked.

Here, the thing being liked is спагетти, and in Russian спагетти is normally treated as plural, so the verb is plural too:

• спагетти нравятся

Compare:

• Мне нравится суп. = I like soup.
• Мне нравятся спагетти. = I like spaghetti.

So the choice between нравится and нравятся depends on the grammatical number of the thing liked.

In standard Russian, спагетти is usually treated as a plural indeclinable noun.

That means:

• it looks the same in different cases
• but it normally takes plural agreement

So:

• вкусные спагетти
• спагетти нравятся
• ем спагетти

The form does not change, but the grammar around it often shows that it is plural.

The dash shows that some words are being left out because they are understood from the first part.

The full version would be something like:

• Дочке нравятся спагетти с сыром, а мне нравятся спагетти с овощным соусом.

Russian very often omits repeated words when they are obvious. The dash marks that omission clearly.

So а мне — с овощным соусом means:

• а мне нравятся спагетти с овощным соусом

This is very natural and common.

А often connects two parts that are being contrasted or compared.

Here the idea is:

• the daughter prefers one kind
• I prefer another kind

So а is a very natural choice.

It is not exactly the same as but every time, and not exactly the same as and either. In this sentence it has a sense like:

• whereas
• while
• and as for me

So а fits the contrast between two people's preferences.

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

So:

• сыр → с сыром
• соус → с соусом

And the adjective must match the noun:

• овощной соус
• с овощным соусом

This is a very common pattern in Russian:

• чай с сахаром
• хлеб с маслом
• макароны с сыром

Because the adjective has to agree with соусом, which is:

• masculine
• singular
• instrumental

So:

• nominative: овощной соус
• instrumental: с овощным соусом

The ending changes because the case changes.

Both forms are possible, but they are different in style.

• дочери is the more standard, neutral dative form of дочь
• дочке is also very common and sounds more conversational, warmer, and more natural in everyday speech

So:

• дочери нравятся спагетти
• дочке нравятся спагетти

Both are correct. In spoken Russian, дочке is extremely common.

Yes. Russian word order is flexible, though each version gives a slightly different emphasis.

The sentence as written:

• Дочке нравятся спагетти с сыром, а мне — с овощным соусом.

puts the people first, which highlights the contrast between the daughter and me.

You could also say:

• Спагетти с сыром нравятся дочке, а с овощным соусом — мне.

This is also correct, but the focus shifts a bit toward the kinds of spaghetti.

So the original word order is natural because it sets up the comparison between two people's tastes very clearly.