Капуста варится быстрее, чем свёкла, а укроп и петрушка делают суп вкуснее.

Because the verb here is вариться, not варить.

• варить = to boil / cook something
• вариться = to boil / cook (about the food itself)

So:

• Я варю суп = I am cooking soup
• Суп варится = The soup is cooking

In your sentence, капуста is the thing undergoing the cooking, so Russian uses варится.

The ending -ся often gives this kind of middle/passive-like meaning: the subject is not doing the action to something else; it is the thing affected by the action.

Russian uses the imperfective here because the sentence expresses a general fact about how ingredients behave.

• Капуста варится быстрее, чем свёкла = Cabbage cooks faster than beetroot
  This is a general statement.

If you say сварится, that is the perfective verb свариться, and in the present-form shape it actually refers to the future:

• Капуста сварится быстрее, чем свёкла = The cabbage will finish cooking faster than the beetroot

So:

• варится = general process / repeated truth / what happens in general
• сварится = one completed future result

Быстрее is the comparative form meaning faster / more quickly.

It comes from быстрый (fast) and is also used adverbially, so with a verb it often means more quickly:

• быстро = quickly
• быстрее = more quickly / faster

In this sentence, it describes how the cabbage cooks:

• Капуста варится быстрее = Cabbage cooks faster

A useful point: Russian comparatives like быстрее do not change for gender, number, or case.

After a comparison with чем (than), the noun is usually in the nominative:

• быстрее, чем свёкла = faster than beetroot

So свёкла stays in its basic dictionary form.

There is also another possible pattern in Russian, without чем, where the second item can appear in the genitive:

• Капуста варится быстрее свёклы

That is also grammatical, but чем + nominative is very common and usually the clearest pattern for learners.

Here а introduces a contrast, but not a strong contradiction.

• но = but
• а = and / while / whereas / but depending on context

In this sentence, а is closer to while or whereas:

• Капуста варится быстрее, чем свёкла, а укроп и петрушка делают суп вкуснее.

This is not saying the second clause opposes the first one. It is just shifting to another related point: cabbage vs beetroot in cooking time, and dill and parsley in flavor.

So а is a very natural connector here.

Because укроп и петрушка is a compound subject: dill and parsley.

Even though each noun is singular by itself, together they form a plural subject, so the verb is plural:

• укроп делает = dill makes
• петрушка делает = parsley makes
• укроп и петрушка делают = dill and parsley make

This works just like English: Dill and parsley make the soup tastier.

Also, Russian often uses singular forms like укроп and петрушка when talking about herbs as substances or ingredients in a general way.

It is in the object case: accusative.

The verb делать takes a direct object:

• делать что? = to make what?

So суп here is accusative. But for many inanimate masculine nouns, the accusative singular looks exactly the same as the nominative singular.

So:

• nominative: суп
• accusative: суп

That is why the form does not visibly change.

This is a very common Russian pattern:

делать + object + comparative

So:

• делают суп вкуснее = make the soup tastier

Here:

• делают = make
• суп = the soup
• вкуснее = tastier

The comparative вкуснее functions as a kind of result/predicative complement: it tells you what the soup becomes.

Compare:

• Этот соус делает мясо вкуснее. = This sauce makes the meat tastier.
• Соль делает суп лучше. = Salt makes the soup better.

You could also say более вкусным in some contexts:

• делают суп более вкусным

But делают суп вкуснее sounds very natural and concise.

Because Russian comparative forms like вкуснее are indeclinable: they do not agree in gender, number, or case.

So you do not change вкуснее to match суп.

Compare:

• суп вкуснее
• каша вкуснее
• блюда вкуснее

The comparative stays the same.

This is different from ordinary adjectives, which do agree:

• вкусный суп
• вкусная каша
• вкусные блюда

But once you use the comparative form, it no longer changes like a regular adjective.

Yes, several.

• капу́ста
• ва́рится
• быстре́е
• свёкла
• укро́п
• петру́шка
• вкусне́е

A very important point is ё in свёкла:

• ё is pronounced like yo
• it is always stressed

So свёкла is pronounced roughly SVYO-kla, not as if it had plain е.

Also, in many printed texts Russians sometimes write е instead of ё, so you may see свекла, but the normal pronunciation is still свёкла.

Not completely. Russian word order is flexible, and changes often affect emphasis rather than basic meaning.

The version you have is quite natural and neutral:

• Капуста варится быстрее, чем свёкла, а укроп и петрушка делают суп вкуснее.

You could rearrange parts of it, but the focus would shift. For example:

• Суп укроп и петрушка делают вкуснее
  This puts extra focus on суп.

• Быстрее, чем свёкла, варится капуста
  This sounds marked and emphasizes the comparison.

So the original order is a good standard model for learners:
subject + verb + comparative, then а + second clause.