Брат любит омлет с грибами, а сестра — кашу с персиком.

Because both words are direct objects of любит, so they must be in the accusative case.

• омлет is a masculine inanimate noun, and for masculine inanimate nouns, the accusative singular is usually the same as the nominative singular.
  • омлет → омлет
• каша is a feminine noun ending in -а, and its accusative singular changes to -у.
  • каша → кашу

So:

• любит омлет
• любит кашу

are both exactly what Russian grammar requires.

Because they come after the preposition с, meaning with, and in this meaning с normally requires the instrumental case.

So:

• грибы → грибами = with mushrooms
• персик → персиком = with peach

This is a very common pattern in Russian:

• чай с лимоном = tea with lemon
• хлеб с маслом = bread with butter
• салат с помидорами = salad with tomatoes

Here с means with, but in a food context it often means made with, containing, or served with.

So:

• омлет с грибами = an omelet with mushrooms / mushroom omelet
• каша с персиком = porridge with peach

It does not necessarily mean the foods are physically separate on the plate. It often describes the ingredients or accompaniment.

The dash shows that the verb любит is left out because it is understood from the first clause.

The full version would be:

Брат любит омлет с грибами, а сестра любит кашу с персиком.

Russian often omits repeated words when the meaning is clear. The dash marks that omission in writing.

So the sentence is basically:

• Brother likes mushroom omelet, and sister likes porridge with peach.

In speech, you usually make a slight pause where the dash is.

А is very common when Russian compares or contrasts two parallel ideas.

Here it means something like:

• whereas
• while
• and as for

So:

• Брат любит омлет с грибами, а сестра — кашу с персиком.

has the sense of:

• Brother likes omelet with mushrooms, whereas sister likes porridge with peach.

Comparison:

• и = and, simple addition
• но = but, stronger contradiction
• а = contrast or comparison between two things

In this sentence, а is the most natural choice.

They are in the nominative case because they are the subjects of the clauses.

• Брат loves/likes...
• сестра loves/likes...

Even though the verb is omitted in the second part, сестра is still the subject of the implied любит.

So both nouns stay in the nominative:

• брат
• сестра

Because its subject, брат, is singular.

Любит is the 3rd person singular form of любить.

• я люблю = I like/love
• ты любишь = you like/love
• он / она любит = he / she likes/loves

In the second clause, любит is omitted, but it is still understood:

• сестра [любит] кашу с персиком

Since сестра is also singular, the same singular verb form fits there too.

No, Russian word order is fairly flexible, but this sentence uses the most neutral, standard order:

subject + verb + object

• Брат любит омлет с грибами
• сестра — кашу с персиком

This is the most natural order if you are just stating facts.

You can change the order for emphasis, but that changes the feel:

• Кашу с персиком любит сестра = It is the porridge with peach that sister likes.
• Омлет с грибами любит брат = It is the mushroom omelet that brother likes.

So the original sentence is the normal, unmarked version.

Because Russian has no articles.

Russian does not have direct equivalents of English a/an and the. Whether something is definite or indefinite is usually understood from context.

So:

• брат can mean the brother or a brother
• сестра can mean the sister or a sister
• омлет can mean an omelet or the omelet

The translation depends on the situation, not on a separate article word.

Yes, absolutely.

You can say:

Брат любит омлет с грибами, а сестра любит кашу с персиком.

That is completely grammatical.

Using the dash is just a more concise, natural way to avoid repeating the same verb. Russian does this often when two clauses have parallel structure.

So both are correct:

• а сестра любит кашу с персиком
• а сестра — кашу с персиком

The version with the dash sounds a bit neater and more elegant in writing.

Usually it means an omelet with mushrooms as part of the dish, but like English with mushrooms, it can sometimes be a little broad.

In most everyday contexts:

• омлет с грибами = mushroom omelet / omelet containing mushrooms
• каша с персиком = porridge with peach added to it or served as a topping

So the phrase names the dish in a natural Russian way, just like English food phrases with with.