Learn Hardcore Russian: Я добавляю морковь в суп, чтобы улучшить вкус. - I add carrots to the soup to improve the taste.

It’s accusative singular because it’s the direct object of добавляю (I add what? → морковь). For many feminine nouns ending in -ь, nominative and accusative look the same: морковь → морковь.

Why do we say в суп and not в супе?

в + accusative (here: в суп) is used for movement/putting something into something (into the soup).
в + prepositional (would be в супе) is used for location (in the soup, where something already is).
So добавляю морковь в суп = I add carrot into the soup.

What case is суп in в суп, and why?

It’s accusative singular: суп (masc. inanimate) has the same form in nominative and accusative, so you don’t see a change. The key is the preposition в with a meaning of “into,” which requires accusative.

Why is there a comma before чтобы?

Because чтобы улучшить вкус is a purpose clause (a dependent clause expressing purpose). In Russian, dependent clauses are normally separated by a comma:
..., чтобы ... = ..., in order to ....

Does чтобы always mean in order to? Can it also mean so that?

Yes. чтобы often means in order to when followed by an infinitive: чтобы улучшить.
It can also mean so that with a finite verb: чтобы суп был вкуснее (so that the soup would be tastier).

Why is it улучшить (perfective) and not улучшать (imperfective) after чтобы?

After чтобы, Russian often uses the perfective infinitive to show a goal/result: you add carrot to achieve a better taste → улучшить (reach an improved state).
улучшать would emphasize an ongoing process of improving (less goal-focused), and is less typical here unless you specifically mean “to keep improving little by little.”

Why is добавляю imperfective present tense—could it be добавлю?

добавляю (imperfective) commonly describes a habitual, repeated, or currently-in-progress action: I add (as a practice / right now).
добавлю (perfective future) would mean I will add (once, as a completed action).

Is Я required, or can it be omitted?

It can be omitted. Russian often drops subject pronouns because the verb ending already shows the person:
(Я) добавляю морковь в суп...
You keep Я if you want emphasis/contrast (e.g., I add it, not someone else).

Can I change the word order? What sounds natural?

Yes, word order is flexible. Neutral is close to what you have:
Я добавляю морковь в суп, чтобы улучшить вкус.
Other natural options shift emphasis:

Чтобы улучшить вкус, я добавляю морковь в суп. (emphasizes purpose)
Морковь я добавляю в суп, чтобы улучшить вкус. (emphasizes carrot)

Why is it улучшить вкус (accusative) and not something like улучшить вкуса?

Because улучшить takes a direct object in the accusative: улучшить что? вкус.
Genitive (вкуса) would be used in different constructions, not with улучшить as a normal transitive verb.

What’s the difference between вкус and вкусно here?

вкус is a noun: taste / flavor → улучшить вкус (improve the taste).
вкусно is an adverb/predicative meaning tasty → you’d rephrase: чтобы суп был вкуснее (so the soup would be tastier).

Can I say морковку instead of морковь?

Yes. морковку is a very common colloquial form (often a diminutive/“everyday” variant). It’s still accusative singular:
Я добавляю морковку в суп...
морковь sounds more neutral/formal.

How is this sentence pronounced (stress), and what happens to в in в суп?

Typical stress: Я доба́вляю морко́вь в суп, что́бы улучи́ть вкус.
In fast speech, в before с often devoices/assimilates, so в суп can sound close to [фсуп] (rather than a clear [в суп]).