Извините, нет ли у Вас ручки?

Here Извините means Excuse me, not really I am sorry in the strong apologetic sense.

It is commonly used to get someone’s attention politely before asking a question. In this sentence, it works just like saying Excuse me, do you happen to have a pen?

Grammatically, извините is the formal/plural imperative form of извинить.

Нет ли is a polite, indirect way to ask whether something is available.

Literally, it looks like is there not... ?, but in natural English it corresponds more to something like:

• Would you happen to have a pen?
• You don’t happen to have a pen, do you?

So even though it contains нет meaning no/not, the speaker is not expecting a negative answer. It is just a softer, more courteous way to ask.

Ли is a question particle. It often has no single direct English equivalent.

Its job is to mark a yes/no question, especially in more formal or careful speech. In нет ли, it helps create that polite, tentative tone.

A useful thing to remember is that ли usually comes right after the word it relates to. Here it follows нет, giving нет ли.

Russian usually expresses possession with a structure that is literally closer to at someone there is.

So:

• У Вас есть ручка = literally At you there is a pen
• natural English: You have a pen

In your sentence, the same possession pattern is used inside the polite question:

• нет ли у Вас ручки?

So у Вас means with you / at your place / in your possession, depending on context.

Capitalizing Вы, Вас, Вам is a polite writing convention when addressing one person respectfully.

So Вас with a capital letter shows extra respect, similar to formal you.

Important points:

• Вас and вас mean the same thing grammatically here
• the capital letter does not change pronunciation
• in modern Russian, lowercase is also very common in ordinary writing

So this is mainly a matter of politeness and style.

Because after нет, Russian normally uses the genitive case for the thing that is absent.

So:

• ручка = nominative singular
• ручки = genitive singular

Compare:

• У Вас есть ручка? = Do you have a pen?
  Here ручка is nominative.
• У Вас нет ручки. = You do not have a pen.
  Here ручки is genitive.
• Нет ли у Вас ручки? follows that same нет pattern, so it also uses the genitive.

Yes, it is quite polite and somewhat formal.

That comes from several things together:

• Извините instead of informal извини
• Вас instead of тебя
• the soft, indirect pattern нет ли

This is a very good sentence for speaking to a stranger, an older person, or someone in a formal setting.

A more neutral everyday version would be:

• Извините, у вас есть ручка?

An informal version to a friend would be:

• У тебя есть ручка?

Not literally. Literally, you are only asking whether the person has a pen.

But in real life, people will often understand that you are probably hoping to borrow one.

If you want to make the borrowing idea explicit, you could say something like:

• Извините, можно одолжить ручку?
• Можно мне ручку на минутку?

So your sentence is indirect and polite, and the intended meaning often depends on context.

Yes, you could. It is grammatically correct.

But the tone is a little different:

• Есть ли у Вас ручка? = more straightforward, sometimes more like a neutral information question
• Нет ли у Вас ручки? = softer, more tactful, more like Would you happen to have a pen?

So the difference is mainly one of tone and politeness, not basic meaning.

To one friend or someone you know well, you would usually switch to ты forms:

• Извини, нет ли у тебя ручки?
• У тебя есть ручка?

The second one is probably the most natural everyday informal version.

So the formal/informal contrast is:

• у Вас → formal/polite
• у тебя → informal/singular

That is one of the most important things to notice in this sentence.