Вона каже, що макарони смачніші, коли на столі є не тільки помідор, а й огірок.

Yes, but in Ukrainian макарони is normally treated as a plural noun. It often refers to pasta or macaroni in general, not necessarily to several separate items in the way English macaronis would.

Because it is plural, words that agree with it are also plural:

• макарони
• смачніші

If you want to refer to one single piece of macaroni, you would use макаронина.

Because смачніші agrees with макарони, which is plural.

The comparative form of смачний is:

• masculine: смачніший
• feminine: смачніша
• neuter: смачніше
• plural: смачніші

So here:

• макарони = plural
• therefore смачніші = plural comparative adjective

English does not show this kind of agreement in tastier, but Ukrainian does.

Що introduces a subordinate clause after каже. It works like English that in a sentence such as She says that...

So:

• Вона каже = she says
• що ... = that ...

In English, that is often omitted. In Ukrainian, що is much more commonly kept.

Коли means when, and here it gives a general situation: pasta is tastier when there is not only a tomato but also a cucumber on the table.

You could use якщо in some contexts, but it would sound more like if and make the sentence feel more conditional:

• коли = when / whenever
• якщо = if

So коли is a natural choice when talking about a repeated or general circumstance.

Because на столі shows location, not motion.

Ukrainian uses:

• на
  • locative for being somewhere: on the table
• на
  • accusative for movement onto something: onto the table

So:

• на столі = on the table
• на стіл = onto the table

Here the sentence describes where the tomato and cucumber are, so the locative столі is required.

That is true for many sentences with the verb to be as a copula, but this sentence uses є in an existential sense, meaning there is / there are.

Compare:

• Вона щаслива. = She is happy.
  No є is needed.
• На столі є огірок. = There is a cucumber on the table.
  Here є is very natural.

So in this sentence, є means that those things are present on the table.

This is a fixed paired construction meaning not only ..., but also ...

So:

• не тільки помідор, а й огірок = not only a tomato, but also a cucumber

A useful point:

• а й is the usual second part here
• й is basically a euphonic variant of і
• You may also see не тільки ..., але й ...

All of these are close in meaning, though не тільки ..., а й ... is very common and natural.

Because the sentence is talking about a tomato and a cucumber as individual items on the table.

So the idea is:

• there is not only a tomato
• but also a cucumber

If the speaker meant several of each, Ukrainian would use plurals:

• помідори
• огірки

The singular here does not mean the sentence is wrong or incomplete; it just paints a specific picture.

They are in the nominative because they are the things that exist in the sentence with є.

In Ukrainian existential sentences, the noun after є is commonly in the nominative:

• На столі є помідор.
• У кімнаті є вікно.

So here:

• помідор = nominative singular
• огірок = nominative singular

Each comma has a reason:

• Вона каже, що ...
  A comma is used before the subordinate clause introduced by що.

• ..., коли на столі ...
  A comma is used before the subordinate clause introduced by коли.

• не тільки помідор, а й огірок
  In this paired construction, a comma is placed before the second part а й.

So the punctuation is normal Ukrainian punctuation for subordinate clauses and paired conjunctions.

Yes. Ukrainian word order is fairly flexible, and the order can change for emphasis.

The given version is natural and clear:

• Вона каже, що макарони смачніші, коли на столі є не тільки помідор, а й огірок.

But Ukrainian could move parts around if the speaker wanted a different focus. For example, the clause with коли could come earlier. The meaning would stay similar, but the emphasis would shift.

So the current order is not the only possible one, just a very normal one.