W tej koszuli brakuje guzika, a zamek w kurtce nie działa dobrze.

Because w here means in and refers to a location, so it requires the locative case.

• ta koszula → w tej koszuli
• ta kurtka → w tej kurtce

So both the noun and any matching word like ten/ta/to change form.

Tej is the locative singular feminine form of ta.

It has to match koszuli, because Polish words that describe or point to nouns must agree with them in:

• gender
• number
• case

Since koszula is feminine singular, and after w we need the locative here, ta changes to tej.

Because koszula is in the locative singular after w.

This is a normal pattern for many feminine nouns ending in -a:

• koszula → w koszuli
• kurtka → w kurtce
• szafa → w szafie

So koszuli does not mean a different word; it is just the case form of koszula.

Because the verb brakować / brakuje usually takes the thing that is missing in the genitive case.

So:

• guzik = button
• guzika = of a button / a button, after brakuje

Compare:

• Brakuje czasu. = Time is lacking.
• Brakuje pieniędzy. = Money is lacking.
• Brakuje guzika. = A button is missing.

A useful way to think about brakuje is there is a lack of... or ...is missing.

So:

• W tej koszuli brakuje guzika literally feels like
  In this shirt, there is a lack of a button

In natural English, we usually say:

• This shirt is missing a button
• A button is missing from this shirt

Polish often structures this idea differently from English.

Yes, absolutely. That is natural Polish too.

The difference is roughly this:

• nie ma guzika = there is no button
• brakuje guzika = a button is missing / should be there but is absent

So brakuje often gives a stronger sense that something expected is missing.

A often means and, but in many sentences it has a slight contrastive feel, like:

• and
• while
• whereas
• sometimes almost but

Here it links two related observations:

• in the shirt, a button is missing
• in the jacket, the zipper does not work well

So a is very natural here because the second part adds a contrasting parallel detail.

Yes, zamek has several meanings, including:

• castle
• lock
• zipper

Polish uses the same word for all of these, and the context tells you which meaning is intended.

Here, because we have w kurtce and we are talking about clothing, zamek clearly means zipper.

For the same reason as w tej koszuli: w with a location uses the locative.

• kurtka → w kurtce

This particular change is common for feminine nouns ending in -ka.

Because dobrze is an adverb, and it describes how the zipper works.

• dobry = good (adjective, used with nouns)
• dobrze = well (adverb, used with verbs)

Since działa is a verb, Polish needs the adverb:

• Zamek działa dobrze. = The zipper works well.

Yes. Polish word order is fairly flexible.

For example, these are all possible:

• W tej koszuli brakuje guzika, a zamek w kurtce nie działa dobrze.
• Zamek w kurtce nie działa dobrze, a w tej koszuli brakuje guzika.
• W kurtce zamek nie działa dobrze, a w tej koszuli brakuje guzika.

The original version sounds natural because it introduces each clothing item first, then says what the problem is.

Because Polish has no articles like English a/an/the.

So Polish simply says:

• koszula
• guzik
• kurtka
• zamek

Whether English needs a or the depends on context and translation, not on a separate Polish word.