Zanim ugotuję zupę, muszę jeszcze obrać marchewkę i znaleźć tarkę.

Because ugotuję comes from the perfective verb ugotować. In Polish, perfective verbs do not have a true present tense. Their present-looking forms are understood as future.

So:

• gotuję = I am cooking / I cook from imperfective gotować
• ugotuję = I will cook / I will have cooked from perfective ugotować

Here, Zanim ugotuję zupę... means Before I cook the soup... or more literally Before I finish cooking the soup...

This is a classic Polish aspect pair:

• gotować = to cook, focusing on the process, repetition, or ongoing action
• ugotować = to cook completely / to cook successfully, focusing on the completed result

So:

• Gotuję zupę. = I’m cooking soup.
• Ugotuję zupę. = I’ll cook the soup / I’ll get the soup cooked.

In your sentence, the speaker is talking about something that will be completed later, so ugotuję is natural.

Zanim means before when it introduces a whole clause with a verb.

Here you have:

• Zanim ugotuję zupę = Before I cook the soup

You cannot use przed in exactly the same way here, because przed is a preposition, and it normally goes before a noun phrase, not a full clause.

Compare:

• Zanim ugotuję zupę... = Before I cook the soup...
• Przed gotowaniem zupy... = Before cooking the soup...

So zanim is the right choice for before + subject + verb.

They are in the accusative case because they are direct objects of the verbs:

• ugotuję zupę = cook soup
• obrać marchewkę = peel a carrot
• znaleźć tarkę = find a grater

All three nouns are feminine singular nouns that change from -a in the nominative to -ę in the accusative:

• zupa → zupę
• marchewka → marchewkę
• tarka → tarkę

This is a very common pattern in Polish.

Because Polish does not use a separate word like English to in this structure.

After musieć = must / have to, Polish simply uses the infinitive:

• muszę obrać = I have to peel
• muszę znaleźć = I have to find

So the pattern is:

• muszę + infinitive

There is no extra word corresponding to English to.

Here jeszcze means something like still, yet, or one more thing remains.

So:

• muszę jeszcze obrać marchewkę i znaleźć tarkę

means:

• I still have to peel the carrot and find the grater
• I have yet to peel the carrot and find the grater

It suggests these tasks are not done yet, but they need to be done before cooking the soup.

Because Polish often omits subject pronouns when they are already clear from the verb ending.

Here:

• ugotuję = I will cook
• muszę = I must / I have to

Both verb forms already show first person singular, so ja is unnecessary.

You could say Ja muszę..., but that would usually add emphasis, for example:

• Ja muszę jeszcze obrać marchewkę = I have to peel the carrot
  perhaps contrasting with someone else

In neutral sentences, Polish usually leaves ja out.

Yes, both are perfective:

• obrać = to peel completely

• znaleźć = to find

  They fit well here because the speaker is talking about completed tasks that need to happen before cooking the soup.

The idea is not just be peeling or be looking for, but rather:

• peel the carrot successfully
• find the grater successfully

If you wanted to focus on process or repeated action, you might use imperfective verbs instead in other contexts, such as:

• obierać = to be peeling / to peel habitually
• znajdować = to be finding / to find repeatedly

But in this sentence, the completed-result idea is the natural one.

Polish has no articles, so there is no separate word for a, an, or the.

That means forms like:

• zupę
• marchewkę
• tarkę

can mean:

• soup / the soup
• a carrot / the carrot
• a grater / the grater

The exact meaning depends on context.

So English has to choose an article when translating, but Polish usually does not mark that difference directly.

Because Zanim ugotuję zupę is a subordinate clause, and in Polish subordinate clauses are normally separated by a comma from the main clause.

So the structure is:

• Zanim ugotuję zupę, = subordinate clause
• muszę jeszcze obrać marchewkę i znaleźć tarkę. = main clause

This comma is standard Polish punctuation.

Yes. Polish word order is more flexible than English, because case endings help show what each word is doing.

The version you have is very natural and neutral:

• Zanim ugotuję zupę, muszę jeszcze obrać marchewkę i znaleźć tarkę.

But other orders are possible, for example:

• Muszę jeszcze obrać marchewkę i znaleźć tarkę, zanim ugotuję zupę.

That means essentially the same thing, though the emphasis shifts a bit. The original version sounds very natural because it first sets the time frame before I cook the soup, then gives the tasks that must be done.

Because one modal verb can govern more than one infinitive.

So:

• muszę obrać marchewkę i znaleźć tarkę

means:

• I have to peel the carrot and find the grater

You do not need to repeat muszę before znaleźć, though you could for emphasis:

• muszę jeszcze obrać marchewkę i muszę znaleźć tarkę

That sounds more emphatic or heavy. The version with one muszę is smoother and more natural.