Запах свіжої кави чудовий.

Because after запах Ukrainian normally uses the genitive case to mean the smell of something.

• кава = the dictionary form, nominative singular
• кави = genitive singular

So:

• запах кави = the smell of coffee

This is a very common pattern in Ukrainian.

Because свіжої has to agree with кави, and кави is in the genitive singular.

The base form is:

• свіжа кава = fresh coffee

But inside this sentence, кава changes to кави, so the adjective changes too:

• свіжа кава → свіжої кави

So both words match in:

• gender: feminine
• number: singular
• case: genitive

Because the endings tell you.

• свіжої is feminine singular genitive
• кави is also feminine singular genitive

But запах is masculine singular nominative, so an adjective describing запах would need a form like свіжий, not свіжої.

So the grammar makes the relationship clear:

• свіжої goes with кави
• чудовий goes with запах

Because it describes запах, and запах is a masculine noun.

In this sentence, чудовий is a predicate adjective: it tells us what the smell is like. It still agrees with the noun it describes:

• запах = masculine singular
• чудовий = masculine singular

If the subject were feminine, the adjective would change too.

Because чудовий is an adjective, and here you are describing a noun: запах.

• чудовий = wonderful, marvelous
• чудово = wonderfully / great / splendidly

So:

• Запах ... чудовий = correct, because запах is a noun
• Запах ... чудово = not correct in this structure

English speakers often confuse adjective and adverb forms here, because English sometimes uses great very broadly.

In Ukrainian, the present-tense form of to be is usually omitted.

So instead of saying something like The smell of fresh coffee is wonderful with an explicit is, Ukrainian normally just says:

• Запах свіжої кави чудовий.

You may see є in some contexts, but in simple present-tense sentences like this, leaving it out is the normal choice.

The sentence has this structure:

• Запах свіжої кави = the subject phrase
• чудовий = the predicate adjective

More literally, it works like:

• [The smell of fresh coffee] [wonderful].

Inside the subject phrase:

• запах = the main noun
• свіжої кави = of fresh coffee

So it is a very typical Ukrainian noun phrase plus adjective-predicate sentence.

Yes. Ukrainian word order is fairly flexible because case endings carry a lot of the grammatical information.

This version is neutral and natural:

• Запах свіжої кави чудовий.

But other versions are possible, for example:

• Чудовий запах свіжої кави.
  This sounds more like A wonderful smell of fresh coffee or puts early focus on wonderful.

• Запах свіжої кави — чудовий.
  With a dash, this can sound more emphatic or contrastive.

So the original order is a good neutral model, but it is not the only possible one.

Because Ukrainian does not have articles.

So nouns appear without separate words for a or the. Whether something is definite or indefinite is understood from:

• context
• word order
• emphasis
• the situation

That is why запах can correspond to the smell or sometimes a smell, depending on context. In this sentence, English usually translates it naturally as the smell.

Запах is usually neutral. It can mean any smell, pleasant or unpleasant.

If you want a more clearly pleasant word, Ukrainian often uses аромат.

So:

• запах кави = smell of coffee
• аромат кави = aroma of coffee

In your sentence, the adjective чудовий makes it clear that the smell is pleasant.