У меня есть кое-какие вопросы.

Russian usually expresses possession with the pattern:

у + person/object in the genitive + есть + thing possessed

So У меня есть кое-какие вопросы literally means something like:

At me there are some questions.

This is the normal, natural way to say I have ... in Russian.

Russian does have the verb иметь (to have), but it is much less common in everyday speech for ordinary possession. It often sounds more formal, bookish, or appears in fixed expressions.

So in normal conversation:

• У меня есть вопрос. = I have a question.
• У меня есть машина. = I have a car.

That is the basic pattern learners should get used to.

Меня is the genitive form of я.

That is because the preposition у requires the genitive case.

So:

• я = I
• у меня = at me / by me

Other examples:

• у тебя = at you / you have
• у него = at him / he has
• у нас = at us / we have

So у меня есть... is built exactly the way Russian possession normally works.

Here есть means there is / there are / exists. It is related to the verb быть (to be) and is used in possession sentences to show existence.

So in this sentence:

• У меня есть вопросы = I have questions / There are questions with me

Yes, Russian also has есть meaning to eat, but that is a different verb form, even though it looks the same.

Context makes the meaning clear:

• Я хочу есть. = I want to eat / I am hungry.
• У меня есть вопросы. = I have questions.

In your sentence, есть clearly means there are / I have.

Sometimes yes, but the meaning or emphasis can shift a little.

With possession, есть is often used when you are simply stating that something exists or is available:

• У меня есть вопрос. = I have a question.
• У меня есть время. = I have time.

If you omit есть, the sentence can sound more conversational, more contextual, or more contrastive depending on the situation:

• У меня вопрос. = I have a question.
• У меня вопросы. = I have questions.

In your sentence, У меня есть кое-какие вопросы sounds very natural because the speaker is introducing the fact that they do, in fact, have some questions.

So:

• with есть = a neutral statement of possession/existence
• without есть = often shorter, more colloquial, sometimes slightly more topic-focused

Кое-какие means something like:

• some
• a few
• certain
• some kind of, but not specified

In this sentence, it suggests I have a few questions or I have some questions.

It often implies that the speaker has specific questions in mind, but is not listing them yet.

So:

• У меня есть кое-какие вопросы. = I have a few questions.

It can sound a little like:

• There are some things I want to ask
• I have a couple of questions

These words can all translate as some, but they are not identical.

кое-какие

This often means some specific ones, though I am not naming them yet.

• У меня есть кое-какие вопросы. = I have a few questions / some particular questions.

This is very natural here.

какие-то

This can mean some or other, some kind of, or sometimes odd / vague / unspecified depending on context.

• У меня есть какие-то вопросы.

This can mean I have some questions, but it may sound vaguer, and in some contexts it can even suggest uncertainty or mild dismissiveness.

некоторые

This means some / certain / several, often in a slightly more neutral or formal way.

• У меня есть некоторые вопросы.

This is understandable, but it can sound more formal or written than кое-какие.

So for natural spoken Russian in this sentence, кое-какие is a very good choice.

Because кое- is a particle/prefix-like element that is written with a hyphen when it forms indefinite pronouns and adjectives.

So you get forms like:

• кое-кто = someone
• кое-что = something
• кое-какой = some kind of / a certain
• кое-какие = some / certain (plural)

That is just the standard spelling rule.

Because кое-какие has to agree with вопросы in number, gender, and case.

Here вопросы is:

• plural
• nominative
• masculine noun in its dictionary form вопрос

So the matching plural nominative form is:

• кое-какие вопросы

Compare:

• кое-какой вопрос = some question / a certain question (singular masculine)
• кое-какие вопросы = some questions (plural)

This is ordinary adjective-like agreement.

In this kind of possession sentence, the thing possessed is normally in the nominative.

So in:

У меня есть кое-какие вопросы

the possessed thing is кое-какие вопросы, and that phrase stays in the nominative plural.

You can think of it as the thing that exists:

• There are some questions
• The questions exist with me / in my possession

That is why it is not in the accusative here.

The basic neutral order is:

У меня есть кое-какие вопросы.

This is the most straightforward way to say it.

But Russian word order is flexible, and changing it can change the emphasis:

• Кое-какие вопросы у меня есть. = I do have some questions. This puts more emphasis on кое-какие вопросы.

• Есть у меня кое-какие вопросы. = I do have some questions, you know. This can sound more expressive or conversational.

So the words can move around, but the original sentence is the most neutral and standard version.

Yes, very natural.

У меня есть кое-какие вопросы is a common, idiomatic way to say that you have a few questions to ask.

It works well in many situations, for example:

• in class
• in a meeting
• in a shop
• when asking for clarification
• when beginning a serious conversation

Depending on tone, it can sound:

• neutral
• polite
• slightly cautious
• a little serious

So this is a very useful sentence pattern to remember.