Если возникнут вопросы, пожалуйста, спрашивайте.

Если means if.

It introduces the condition for the rest of the sentence:

• Если возникнут вопросы = If questions come up
• пожалуйста, спрашивайте = please ask

So the whole structure is basically:

• If X happens, please do Y.

Возникнут is the future tense of возникнуть, which is a perfective verb.

Here it means something like:

• arise
• come up
• occur

So:

• возникнут вопросы = questions arise / questions come up

Why future? Because the sentence is talking about a possible situation that may happen later:

• If questions come up...

Why perfective? Because it refers to the possible occurrence of questions as an event, not an ongoing process.

Compare:

• Если возникают вопросы... = If questions are arising / if questions tend to come up
• Если возникнут вопросы... = If questions come up at some point

In this sentence, the perfective future is the natural choice.

Because the subject is вопросы, which is plural.

Russian verbs in the past and future agree with the subject in number, so:

• вопрос возникнет = a question will arise
• вопросы возникнут = questions will arise

That is why the verb ends in the plural form -ут.

Because вопросы is the grammatical subject of возникнут.

In Russian, the subject of a verb is usually in the nominative case.

So here:

• вопросы = nominative plural of вопрос
• возникнут вопросы = questions will arise

Even though in English we might think of have questions, Russian often expresses this idea differently:

• У вас будут вопросы = You will have questions
• Возникнут вопросы = Questions will arise / come up

Both are possible, but this sentence uses the second pattern.

Because возникнут вопросы is a very natural Russian way to express the idea that questions may appear during discussion, explanation, study, and so on.

It sounds a bit more formal and polished than:

• Если у вас будут вопросы...

Both mean nearly the same thing in context:

• Если возникнут вопросы, пожалуйста, спрашивайте.
• Если у вас будут вопросы, пожалуйста, спрашивайте.

The version with возникнут is common in professional, educational, and polite written or spoken communication.

Спрашивайте is the imperative form of спрашивать.

It means:

• ask
• or more politely, please ask

The ending -йте can show:

1. plural command: speaking to more than one person
2. formal/polite singular: speaking respectfully to one person

So спрашивайте can mean:

• ask to several people
• ask to one person in a polite/formal way

In this sentence, it is a polite invitation.

This is an aspect question.

• спрашивайте = imperfective imperative
• спросите = perfective imperative

In this sentence, спрашивайте is more natural because it gives a general, open-ended invitation:

• If questions come up, feel free to ask.

The imperfective imperative often suggests:

• repeated action
• general permission
• an ongoing invitation

By contrast, спросите can sound more like:

• ask once
• go ahead and ask

So:

• пожалуйста, спрашивайте = please ask / feel free to ask
• пожалуйста, спросите = more like please ask in reference to a specific question or act

Here, the imperfective fits better.

Here пожалуйста means please.

It softens the imperative and makes the sentence polite:

• спрашивайте = ask
• пожалуйста, спрашивайте = please ask

Russian пожалуйста is used very often for politeness, just like please in English.

Because Если возникнут вопросы is a subordinate clause introduced by если.

Russian normally separates this kind of clause from the main clause with a comma:

• Если возникнут вопросы, пожалуйста, спрашивайте.

This is the same basic idea as in English:

• If you have questions, please ask.

Russian punctuation is quite strict here, so the comma is expected.

Because Russian often omits pronouns when they are not necessary.

In English, you often need you:

• If you have questions, please ask.

In Russian, the meaning is clear from context and from the verb form:

• спрашивайте already implies you plural/formal
• возникнут вопросы naturally implies that the questions come up for the listener

If you wanted, you could make it more explicit:

• Если у вас возникнут вопросы, пожалуйста, спрашивайте.

But the shorter version is completely natural.

It is polite and somewhat formal/neutral.

Why?

• возникнут вопросы sounds fairly polished
• спрашивайте with -йте is polite or plural
• пожалуйста adds politeness

You might hear it from:

• a teacher
• a customer support person
• a lecturer
• a colleague in a professional setting

A more informal version to one friend would be:

• Если будут вопросы, спрашивай.

A polite version to one person or several people is:

• Если возникнут вопросы, пожалуйста, спрашивайте.

Yes, absolutely.

That version is also very natural and common:

• Если будут вопросы, пожалуйста, спрашивайте.

Compared with Если возникнут вопросы:

• Если будут вопросы = slightly simpler, more direct
• Если возникнут вопросы = slightly more formal, a bit more elegant

Both are correct, and both are widely used.