Проверь, пожалуйста, есть ли у нас молоко.

Проверь is the informal singular imperative of проверить.

So it means a direct instruction to one person: check.

• Проверь = tell one person you know well, a friend, a child, etc.
• Проверьте = formal singular or plural

Russian imperatives often omit the subject pronoun, so ты is understood here but not stated.

Because проверить is perfective, and here the speaker wants one complete result: do the check and find out.

• Проверь = check it once and get the answer
• Проверяй = keep checking / check regularly / be in the process of checking

In this sentence, the idea is not an ongoing activity. It is a single completed action, so проверь is the natural choice.

Here пожалуйста is used as a parenthetical politeness word, similar to please.

That is why it is set off with commas:

Проверь, пожалуйста, ...

You will very often see пожалуйста separated this way when it is not grammatically part of the sentence structure, but simply adds politeness.

Compare:

• Проверь, пожалуйста, ... = Please check ...
• Пожалуйста, проверь ... = also possible, with пожалуйста at the beginning

Есть ли introduces an indirect yes/no question, meaning whether / if there is.

So:

• есть ли у нас молоко = whether we have milk / if there is milk

A direct question would be:

• У нас есть молоко? = Do we have milk?

But after Проверь you need an embedded question, so Russian uses ли:

• Проверь, есть ли у нас молоко.

The particle ли usually comes after the word it relates to most directly, often near the beginning of the clause.

In есть ли у нас молоко, the focus is on the existence of the milk:

• есть ли = whether there is / whether we have

That is why ли follows есть.

Russian word order can sometimes shift for emphasis, but есть ли у нас молоко is the normal neutral order here.

Russian usually expresses possession with у + genitive plus есть.

So instead of saying we have milk with a direct equivalent of have, Russian normally says something closer to:

• у нас есть молоко
• literally: at us there is milk

Here:

• у = at
• нас = genitive form of мы

This is one of the most important basic Russian patterns for possession.

Because the preposition у requires the genitive case.

The pronoun мы changes like this:

• nominative: мы
• genitive: нас

So:

• мы = we
• у нас = at us / we have

That is why у мы is impossible, and у нас is correct.

Молоко is in the nominative singular.

In this kind of existential sentence, the thing whose presence or existence is being discussed is normally in the nominative:

• У нас есть молоко.
• Есть ли у нас молоко?

Here the sentence is asking whether milk is present/available, so молоко stays in its dictionary form.

An English speaker may expect something like an object case after have, but Russian is not building the idea the same way. It is really closer to is there milk at our place?

Usually, no in this sentence.

Russian often omits the present tense of to be in sentences like:

• Он врач. = He is a doctor.

But есть here is not just a missing copula. It has an existential meaning: there is / there exists / is available.

So:

• У нас есть молоко. = We have milk / There is milk at our place.

Without есть, the sentence would not sound like the normal neutral way to ask whether milk is available.

It is not really missing; it is just normally omitted.

Russian usually leaves out personal pronouns when the verb form already makes the person clear. The imperative Проверь already tells you that the speaker is addressing one person informally.

So:

• Проверь, пожалуйста... is natural
• Ты проверь, пожалуйста... is possible, but it adds extra emphasis

Change Проверь to Проверьте.

So:

• Проверь, пожалуйста, есть ли у нас молоко. = informal, one person
• Проверьте, пожалуйста, есть ли у нас молоко. = polite to one person or addressed to several people

Everything else stays the same.

Yes, a little. Russian word order is flexible, though the original sentence is the most neutral and natural.

Possible variants include:

• Пожалуйста, проверь, есть ли у нас молоко.
• Проверь, есть ли у нас молоко, пожалуйста.

These all mean basically the same thing, but the emphasis and rhythm change slightly.

The original version:

• Проверь, пожалуйста, есть ли у нас молоко.

is a very standard, polite, neutral way to say it.

In normal usage, it means check whether we have any milk available.

With mass nouns like молоко, Russian often uses the bare noun in this kind of sentence without needing a separate word for any.

So the idea is naturally understood as:

• Is there milk?
• Do we have milk?
• Is there any milk at home / with us?

The exact nuance comes from context, but in everyday speech this is the normal way to ask.