Морква й буряк дорогі.

Because together they make a compound subject: морква й буряк = carrots and beetroot.

In Ukrainian, when two singular nouns are joined by і / й (and), the predicate usually goes in the plural:

• Морква й буряк дорогі. = Carrots and beetroot are expensive.

So дорогі agrees with the whole pair, not with just one noun.

Й and і both mean and here. Ukrainian often switches between them for euphony—to make the sentence sound smoother.

In Морква й буряк, й is used because it flows well between the vowels/consonants in that phrase.

You can think of it as similar to choosing a smoother-sounding form in speech. In many cases:

• і = and
• й = and

The meaning does not change here.

Yes, but in many simple sentences they can all mean and.

• і = the basic word for and
• й = a euphonic variant of і
• та = also means and, often a little more formal or stylistically different

So these are all possible in many contexts:

• Морква і буряк дорогі.
• Морква й буряк дорогі.
• Морква та буряк дорогі.

Your sentence uses й because it sounds natural.

They are in the nominative case.

That is because they are the subject of the sentence—the things being talked about:

• морква = carrot
• буряк = beetroot
• морква й буряк = carrot and beetroot

The adjective дорогі is also in the nominative plural because it agrees with that subject.

In Ukrainian, the present-tense form of to be is usually omitted in sentences like this.

So:

• Морква й буряк дорогі.

literally looks like:

• Carrot and beetroot expensive.

But it naturally means:

• Carrot and beetroot are expensive.

This is completely normal in Ukrainian.

• морква is feminine
• буряк is masculine

Even though the nouns have different genders, the adjective becomes plural when they are joined together:

• морква
  • буряк → дорогі

So you do not use feminine singular дорога or masculine singular дорогий here.

Because дорогий and дорога are singular forms:

• дорогий = masculine singular
• дорога = feminine singular
• дорогі = plural

Since the sentence has two items joined by and, Ukrainian uses the plural adjective:

• морква дорога = the carrot is expensive
• буряк дорогий = the beetroot is expensive
• морква й буряк дорогі = the carrot and beetroot are expensive

No. Дорогий can mean either:

• expensive
• dear / beloved

In this sentence, it clearly means expensive, because it is talking about vegetables and their price:

• Морква й буряк дорогі. = Carrots and beetroot are expensive.

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

The neutral version is:

• Морква й буряк дорогі.

But other orders are possible for emphasis, for example:

• Дорогі морква й буряк.

That could sound more marked or literary, with extra focus on expensive.

For a learner, the given order is the safest and most natural to produce.

A helpful approximation is:

• морква → mor-KVA
• й → like y in yes
• буряк → boo-RYAK
• дорогі → do-ro-HI

Stress is important in Ukrainian:

• морква́
• буря́к
• дорогі́

So the full sentence sounds roughly like:

• mor-KVA y boo-RYAK do-ro-HI

In most contexts, буряк refers to beet / beetroot.

For an English speaker, beetroot is often the clearest translation, especially in British English. In American English, beet is usually enough.

So in this sentence, буряк means the vegetable, not something abstract or unusual.

You would make the adjective singular to match the noun.

For example:

• Морква дорога. = The carrot is expensive.
• Буряк дорогий. = The beetroot is expensive.

But with both together, it becomes plural:

• Морква й буряк дорогі. = The carrot and beetroot are expensive.