Minha irmã prefere fritar o peixe no óleo, mas minha mãe gosta de assar a carne no forno.

Because meu/minha agrees with the thing possessed, not with the person who owns it.

• irmã is feminine singular → minha irmã
• mãe is feminine singular → minha mãe

Compare:

• meu irmão
• meu pai
• minha casa

So minha is used because irmã and mãe are feminine nouns.

Yes. In Brazilian Portuguese, both forms are possible:

• minha irmã
• a minha irmã

In this sentence, minha irmã and minha mãe without the article sound very natural. Adding a is also correct in many regions and styles, and can sound a little more specific or emphatic.

Because these two verbs use different patterns.

• preferir + infinitive / noun
  • prefere fritar
  • prefere café
• gostar de + infinitive / noun
  • gosta de assar
  • gosta de café

So the de is required after gostar, but not after preferir.

After verbs like preferir and gostar de, Portuguese commonly uses the infinitive to express an action.

So:

• prefere fritar
• gosta de assar

This works like English to fry and to roast/bake, but Portuguese just uses the infinitive form directly.

Portuguese often uses the definite article with foods and other nouns in a general or habitual sense.

So these sound very natural:

• fritar o peixe
• assar a carne

If you remove the articles, the sentence can still be understandable in some contexts, but o peixe and a carne are the more natural choices here.

No is a contraction of em + o.

• em + o = no

So:

• no óleo = in the oil
• no forno = in the oven

This happens because óleo and forno are masculine singular nouns.

A similar feminine form is:

• na panela = em + a

You could say com óleo in some situations, but it does not mean exactly the same thing.

• fritar no óleo suggests frying in oil, with oil as the cooking medium
• fritar com óleo just means frying with oil

In this sentence, no óleo is the more natural choice. Likewise, no forno is the normal way to say in the oven.

They are both in the present indicative, third person singular.

• minha irmã prefere
• minha mãe gosta

They are third person singular because the subjects are minha irmã and minha mãe, each one person.

No. They are different words.

• mas = but
• mais = more

In this sentence, mas introduces a contrast:

• Minha irmã prefere... mas minha mãe gosta...

So here it must be mas, not mais.

The accents are important.

• irmã: the ã is nasal
• mãe: the ãe is also nasal
• óleo: the ó shows the stressed syllable and an open o sound

For an English speaker, the biggest thing to notice is that ã and ãe are nasal sounds, not plain a sounds.

Because the subject changes.

The first part is about minha irmã.
The second part is about minha mãe.

When the subject changes, Portuguese normally states the new subject clearly:

• Minha irmã prefere...
• mas minha mãe gosta...

If you left out minha mãe, it would sound like the same person was still being talked about.

They refer to different cooking methods.

• fritar = to fry
• assar = to roast / bake / grill, depending on context

In this sentence:

• fritar o peixe no óleo = frying the fish in oil
• assar a carne no forno = roasting/baking the meat in the oven

So the contrast is not just between two foods, but also between two cooking methods.