Я взяла новый купальник, а брат — плавки и большое полотенце.

Because Russian past-tense verbs agree with the subject in gender and number.

• я взял = I took (said by a male speaker)
• я взяла = I took (said by a female speaker)

So взяла tells us the speaker is female.

Because новый agrees with купальник, not with the speaker.

In Russian, adjectives agree with the noun they describe in gender, number, and case:

• купальник is masculine singular
• so the adjective is новый

Even if the speaker is female, that does not affect the adjective here.

Compare:

• новый купальник = a new swimsuit
• новая сумка = a new bag
• новое полотенце = a new towel

Because купальник is an inanimate masculine noun, and in the accusative singular, many inanimate masculine nouns look exactly like the nominative.

So:

• nominative: новый купальник
• accusative: новый купальник

After взять (to take), the object is normally in the accusative case.

This is different from animate masculine nouns:

• Я вижу нового студента = I see the new student

But with inanimate nouns:

• Я вижу новый дом = I see the new house

Not exactly. А often means something like and, while, or whereas, with a sense of contrast or switching to another topic.

Here it connects two parallel ideas:

• Я взяла новый купальник
• а брат — плавки и большое полотенце

So the sense is:

• I took a new swimsuit, and/while my brother took swimming trunks and a big towel.

It is more contrastive than и.
If you used и, it would sound more like simple addition.

The dash shows that the verb взял is being left out because it is understood from the first part of the sentence.

Full version:

• Я взяла новый купальник, а брат взял плавки и большое полотенце.

Russian often omits repeated words when the meaning is obvious. The dash marks that omission clearly.

So:

• брат — плавки и большое полотенце
  really means
• брат взял плавки и большое полотенце

Yes, but then you would usually repeat the verb:

• Я взяла новый купальник, а брат взял плавки и большое полотенце.

If you omit the verb, the dash is the normal way to show that omission:

• Я взяла новый купальник, а брат — плавки и большое полотенце.

The version with the dash is natural and a bit more compact.

Because плавки is a noun that is normally used in the plural when it means swimming trunks.

This is similar to English words like:

• shorts
• pants
• glasses

Even though it refers to one item, grammatically it is plural.

So:

• плавки = swimming trunks

You would not normally say плавка with this meaning.

Because полотенце is a neuter singular noun.

So the adjective must also be neuter singular:

• masculine: большой
• feminine: большая
• neuter: большое
• plural: большие

That gives:

• большое полотенце = a big towel

Also, since полотенце is inanimate, its accusative singular is the same as its nominative singular here.

They are in the accusative case, because they are the direct objects of взял / взяла.

• взяла купальник
• взял плавки
• взял полотенце

Their forms happen to look like nominative forms:

• плавки: plural inanimate nouns usually have nominative = accusative
• полотенце: neuter singular inanimate nouns also usually have nominative = accusative

So the case is accusative, even though the form does not visibly change.

In Russian, possessive words like my are often omitted when the meaning is obvious from context.

So а брат here naturally means and my brother.

Russian also has no articles like a and the, so context does a lot of work.

If the speaker wanted to be more explicit, they could say:

• а мой брат = and my brother

But in many everyday sentences, just брат sounds perfectly natural.

The basic meaning stays the same, but the chosen word order sounds natural and helps the contrast.

• Я взяла новый купальник, а брат — плавки и большое полотенце.

This presents:

1. what I took
2. what my brother took

Russian word order is more flexible than English, but it is not random. It often reflects topic, focus, and contrast.

This sentence is nicely balanced:

• Я ...
• а брат ...

So it sounds smooth and natural.

Yes. The grammatical gender of the noun does not have to match the real-life gender of the person using it.

Купальник is grammatically masculine, but it can still refer to a swimsuit for a woman.

That is why you get:

• новый купальник

even if the speaker is female and means her swimsuit.

This is normal in Russian: grammatical gender belongs to the word, not necessarily to the owner or user of the object.