Пусть бритва и новая, брат всё равно предпочитает бриться вечером, а не утром.

Not exactly. In Пусть он войдёт, пусть means let: Let him come in.

In your sentence, пусть is being used in a concessive sense, meaning something like:

• although
• even if
• granted that

So Пусть бритва и новая... means Even though the razor is new... or Granted, the razor is new...

This use is fairly common in Russian and often sets up a contrast with the main clause.

This и is part of the concessive pattern. In sentences like this, пусть ... и ... often has the sense of:

• even though ... is ...
• granted that ... is ...

So:

• Пусть бритва и новая = Even though the razor is new

The и does not simply mean and here in the normal additive sense. It helps reinforce the idea of concession.

A rough structural parallel would be:

• Пусть он и устал, он продолжает работать.
• Even though he is tired, he keeps working.

Because бритва is the subject of the clause бритва ... новая.

This part is essentially equivalent to:

• Бритва новая = The razor is new

Since бритва is the subject, it stays in the nominative:

• бритва = nominative singular
• новая agrees with it in gender, number, and case

So the concessive clause is built on a normal subject + predicate relationship: бритва + новая.

Good question. With adjectives in the present tense, Russian usually uses the short form or the full form in nominative, depending on style and meaning.

Here, новая is a full adjective in the nominative because it is describing the subject in a simple equational statement:

• Бритва новая = The razor is new

That is perfectly normal.

You may also see short forms in some contexts, but with many everyday adjectives, especially in modern speech, the full form is very common.

So in this sentence:

• бритва = nominative subject
• новая = nominative predicate adjective agreeing with бритва

Всё равно means all the same, anyway, or still.

In this sentence it shows that the fact mentioned in the first clause does not change the result:

• Пусть бритва и новая, брат всё равно предпочитает...
• Even though the razor is new, brother still prefers...

This pairing is very common in Russian:

• concessive idea first: although / even though
• then всё равно in the main clause: still / anyway

For example:

• Пусть холодно, мы всё равно пойдём гулять.
• Even though it’s cold, we’ll still go for a walk.

Because предпочитать means to prefer, and after prefer Russian usually uses an infinitive if the subject is the same.

So:

• брат предпочитает бриться = the brother prefers to shave

This is the same kind of structure as:

• Он любит читать = He likes to read
• Она хочет спать = She wants to sleep

If Russian used a plain finite verb here, it would mean something different structurally. The infinitive is needed because it expresses the action that he prefers.

Because бриться means to shave oneself, while брить means to shave someone/something.

Compare:

• брить = to shave someone
• бриться = to shave oneself / to shave

Examples:

• Он бреет клиента. = He is shaving a client.
• Он бреется. = He is shaving.

In your sentence, the brother is the one doing the shaving to himself, so Russian uses the reflexive infinitive:

• предпочитает бриться = prefers to shave

Russian often uses the instrumental case to express time in the sense of in the morning, in the evening, during the day, etc.

So:

• утро → утром = in the morning
• вечер → вечером = in the evening

This is a very common adverbial use of the instrumental.

Other examples:

• днём = during the day / in the daytime
• ночью = at night (this one is a bit different historically, but serves a similar function)

So:

• бриться вечером, а не утром
• to shave in the evening, not in the morning

Because а не marks an explicit contrast: in the evening, not in the morning.

Here а is not simply and. It often means whereas, but, or signals contrast between two alternatives.

So:

• вечером, а не утром = in the evening, rather than in the morning

This sounds more natural and contrastive than simply не утром.

Compare:

• Он работает дома, а не в офисе.
• He works at home, not in the office.

The а helps set up the second option as the rejected alternative.

Because Пусть бритва и новая is a separate concessive clause, and it is followed by the main clause:

• Пусть бритва и новая, брат всё равно предпочитает...

Russian normally separates this kind of subordinate or clause-like concessive phrase with a comma.

So the sentence has two parts:

1. Пусть бритва и новая = concessive background: Even though the razor is new
2. брат всё равно предпочитает бриться вечером... = main statement: the brother still prefers to shave in the evening...

The given word order is natural and emphasizes the concession first:

• Пусть бритва и новая, брат всё равно предпочитает бриться вечером...

That means something like: Granted, the razor is new, but...

You could rearrange parts of the sentence, but the emphasis would shift. Russian word order is flexible, yet not random.

For example, the current order highlights this contrast:

• fact conceded: the razor is new
• result unchanged: he still prefers shaving in the evening

Also, placing всё равно before предпочитает is natural because it modifies the whole idea: despite that fact, his preference remains the same.

Russian has no articles, so брат by itself can mean:

• brother
• the brother
• my brother
• his brother

The exact English choice depends on context.

In isolation, an English learner might translate it as:

• his brother still prefers...
• the brother still prefers...
• brother still prefers... only in special styles, not normal standard English

So Russian leaves this unspecified unless context makes it clear. English usually has to choose.