Мой брат любит листать старые фотографии и вспоминать лето.

Because мой брат is the subject of the sentence, so it is in the nominative case.

• мой = my
• брат = brother

In Russian, the subject of a sentence usually appears in the nominative:

• Мой брат любит... = My brother likes...

If brother were the object, its form would change:

• Я вижу моего брата. = I see my brother.

Любит is the 3rd person singular form of the verb любить (to love / to like).

Since the subject is мой брат = my brother = he, Russian uses:

• я люблю = I like
• ты любишь = you like
• он / она любит = he / she likes

So:

• Мой брат любит... = My brother likes...

After любить, Russian often uses an infinitive to express liking to do something.

So the pattern is:

• любить + infinitive

Examples:

• Я люблю читать. = I like to read.
• Она любит гулять. = She likes to walk.
• Мой брат любит листать... и вспоминать... = My brother likes to flip through... and remember...

This is very similar to English likes to do or likes doing.

Листать means to leaf through, to flip through, or to turn the pages of something.

It is not exactly the same as читать (to read).
If someone листает фотографии, they are usually looking through them one after another, often casually.

So:

• листать старые фотографии = to flip through old photographs

This verb is commonly used with:

• книгу = a book
• журнал = a magazine
• альбом = an album
• фотографии = photographs

Because фотографии is the direct object of листать, so it should be in the accusative plural.
But for inanimate plural nouns, the accusative plural is usually the same as the nominative plural.

So:

• nominative plural: старые фотографии
• accusative plural: старые фотографии

That is why the form does not visibly change.

Compare with an animate noun:

• Я вижу старых друзей. = I see old friends.

There, the accusative plural is different because friends are animate.

Because вспоминать takes a direct object in the accusative case.

So Russian says:

• вспоминать лето = to remember the summer
• вспоминать детство = to remember childhood
• вспоминать поездку = to remember the trip

There is no preposition needed here.

Also, лето is a neuter noun, and in the singular its accusative form is the same as its nominative form:

• nominative: лето
• accusative: лето

So the form stays the same.

Because the sentence describes a general habit or preferred activity, not a single completed action.

Imperfective verbs are commonly used for:

• repeated actions
• ongoing actions
• general activities
• things someone likes doing

So here:

• любит листать = likes flipping through
• любит вспоминать = likes remembering

If you used perfective verbs, it would sound more like a single completed act, which does not fit as naturally after любит in this context.

Yes. И means and, and it joins the two infinitives:

• листать старые фотографии
• вспоминать лето

Both depend on любит.

So the structure is:

• Мой брат любит
  • листать старые фотографии
  • и вспоминать лето

In other words, he likes doing both things.

Yes. Russian word order is more flexible than English word order.

The neutral order here is:

• Мой брат любит листать старые фотографии и вспоминать лето.

But other orders are possible for emphasis, for example:

• Старые фотографии мой брат любит листать и вспоминать лето.
• Лето мой брат любит вспоминать, листая старые фотографии.

These may sound more marked or literary depending on context.
For a learner, the original sentence is the most natural and straightforward version.

The stress is:

• мой брат любИт листАть стАрые фотографИи и вспоминАть лЕто

More clearly:

• любИт
• листАть
• стАрые
• фотографИи
• вспоминАть
• лЕто

Stress matters in Russian, so it is worth learning words together with their stressed syllables.

Sometimes yes, but in this sentence like is usually the most natural translation.

Russian любить can mean:

• to love
• to like
• to be fond of

When it is followed by an infinitive, it often means to like doing something:

• люблю читать = I like reading
• любит вспоминать лето = he likes remembering the summer

Translating it as loves to flip through old photographs and remember summer is possible, but likes is more neutral and common.