Meine Nichte lebt vegan, aber ihr Bruder isst trotzdem gern Rührei.

Because Nichte is a feminine noun: die Nichte.

The possessive word mein changes its ending to match the gender, number, and case of the noun that follows it. In the nominative singular feminine, it becomes meine.

So:

• mein Bruder
• meine Schwester
• mein Kind
• meine Freunde

Here, Nichte is the subject of the sentence, so it is nominative, and since it is feminine singular, meine is the correct form.

Both can be possible, but they are not exactly the same in nuance.

• vegan sein = to be vegan
• vegan leben = to live vegan / to follow a vegan lifestyle

In this sentence, lebt vegan emphasizes the person’s way of living, especially their eating habits and lifestyle choices. It is a very natural German expression.

So:

• Sie ist vegan. = She is vegan.
• Sie lebt vegan. = She lives vegan / follows a vegan lifestyle.

English often just says is vegan, but German very commonly uses vegan leben.

Yes, vegan is basically an adjective, but in this sentence it is not standing before a noun. It is being used predicatively / adverbially with lebt.

Compare:

• eine vegane Ernährung → adjective with ending before a noun
• Sie lebt vegan → no ending, because it is not directly before a noun

This is very common in German. Adjectives do not take endings when they are used this way:

• Er ist alt
• Sie fährt schnell
• Wir essen gesund
• Meine Nichte lebt vegan

Yes. Here ihr means her.

It is a possessive determiner referring back to meine Nichte. So ihr Bruder means her brother.

The form ihr changes depending on the noun that follows. Since Bruder is masculine singular nominative, the correct form is ihr.

Compare:

• ihr Bruder = her brother
• ihre Schwester = her sister
• ihr Kind = her child
• ihre Eltern = her parents

A learner may notice that ihr can also mean other things in German, such as to her, their, or your in formal speech, but here the context clearly makes it a possessive meaning her.

Essen is the infinitive, and isst is the 3rd person singular present tense form.

The subject is ihr Bruder, which means he. So the verb must be conjugated like er isst.

Present tense of essen:

• ich esse
• du isst
• er / sie / es isst
• wir essen
• ihr esst
• sie / Sie essen

So ihr Bruder isst means her brother eats.

Because aber is a coordinating conjunction. It connects two main clauses, but it does not force the verb to the end.

The structure is:

• Meine Nichte lebt vegan
• aber ihr Bruder isst trotzdem gern Rührei

In each main clause, the conjugated verb stays in the normal verb-second position.

So after aber, the new clause begins, and that clause still follows normal main-clause word order:

• ihr Bruder = position 1
• isst = position 2

This is different from conjunctions like weil or dass, which send the verb to the end.

Compare:

• ..., aber ihr Bruder isst gern Rührei.
• ..., weil ihr Bruder gern Rührei isst.

Trotzdem means nevertheless, all the same, or still.

It shows a contrast: the niece lives vegan, but her brother still likes to eat scrambled eggs. In other words, her vegan lifestyle does not mean he follows the same pattern.

It adds a stronger sense of contrast than aber alone.

Compare:

• ..., aber ihr Bruder isst gern Rührei.
  = simple contrast
• ..., aber ihr Bruder isst trotzdem gern Rührei.
  = stronger idea of even so / despite that

Because in a normal German main clause, the conjugated verb comes early, and adverbs like trotzdem often appear in the middle field of the sentence.

So:

• ihr Bruder = subject
• isst = conjugated verb
• trotzdem = adverb
• gern = adverb
• Rührei = object

That word order is very natural.

You can sometimes move adverbs around for emphasis, but isst trotzdem gern Rührei is a very standard and idiomatic order.

German often expresses liking to do something with the adverb gern rather than with a separate verb like to like.

So:

• Er isst gern Rührei = He likes eating scrambled eggs / He enjoys eating scrambled eggs

This is one of the most important patterns in German.

Compare:

• Ich trinke gern Kaffee. = I like drinking coffee.
• Sie liest gern. = She likes reading.
• Wir reisen gern. = We like traveling.

German does also have the verb mögen, but gern is often the most natural choice when talking about enjoying an activity.

In everyday modern German, gern and gerne usually mean the same thing.

So both of these are fine:

• Er isst gern Rührei.
• Er isst gerne Rührei.

Gern is just a slightly shorter form and is extremely common. Learners can treat them as interchangeable in most situations.

Because Rührei is being used as a mass noun / food item in a general sense.

German often leaves out the article with food, drinks, and other uncountable or general substances when speaking generally:

• Ich trinke Kaffee.
• Sie isst Brot.
• Er kocht Reis.
• Ihr Bruder isst gern Rührei.

You could use an article in some contexts, but that would usually sound more specific:

• Er isst das Rührei. = He is eating the scrambled eggs.
• Er bestellt ein Rührei. = He orders a scrambled egg dish / a portion of scrambled eggs.

Here, the sentence is talking about what he likes in general, so no article is natural.

It is in the accusative case, because it is the direct object of isst.

He eats what?
→ Rührei

For this particular noun, you cannot see the case from the noun form alone, because Rührei itself does not change here. But grammatically it is accusative.

Also, Rührei is a neuter noun: das Rührei.

Because aber is connecting two main clauses, and in German these are normally separated by a comma.

So:

• Meine Nichte lebt vegan, aber ihr Bruder isst trotzdem gern Rührei.

This is standard German punctuation.

In isolation, ihr can indeed mean several things, including her, their, or your in formal language. But in this sentence, the context makes her the natural reading.

Why?

Because the first clause introduces meine Nichte, a singular female person. The second clause then refers back to that person with ihr Bruder = her brother.

If it meant their brother, the sentence would need a plural group as the reference point, which it does not have here.

Yes, you can say:

Meine Nichte ist vegan, aber ihr Bruder isst trotzdem gern Rührei.

That is also correct and natural.

The difference is mainly nuance:

• ist vegan = describes her status or identity
• lebt vegan = emphasizes her lifestyle and habits

In many contexts, both work well.