На празднике будет около десяти гостей.

Because на празднике answers the question where? — at the party / at the celebration.

After на when it means location, Russian uses the prepositional case:

• на празднике = at the party

By contrast, на праздник would usually mean for the party or to the party depending on context, not at the party.

So here:

• На празднике будет... = At the party, there will be...

With many events, Russian idiomatically uses на to mean at:

• на концерте = at the concert
• на уроке = at the lesson/class
• на встрече = at the meeting
• на празднике = at the celebration/party

So this is mostly something you learn as a normal collocation:

• быть на празднике = to be at the party

В празднике would sound more like in the celebration in a more abstract sense, not the normal way to say someone is physically attending the event.

Here будет means there will be.

It is the future-tense form of быть (to be). In Russian, sentences of existence often use forms of быть where English uses there is / there are / there will be.

So:

• На празднике будет около десяти гостей.
• Literally: At the party will be about ten guests.
• Natural English: There will be about ten guests at the party.

Russian does not need a separate dummy word like English there.

This is very common in Russian with numbers and quantity expressions.

When Russian presents something as a quantity existing somewhere, it often uses singular будет:

• будет пять человек
• будет много людей
• будет около десяти гостей

The idea is not so much the guests will do something, but there will be a quantity of guests. Russian often treats that whole quantity as one unit.

So in this sentence, будет is the normal choice.

Because около requires the genitive case.

The basic number is:

• десять = ten

But after около (about / around), it changes to genitive:

• около десяти = about ten

This is a very important pattern:

• около часа = about an hour
• около дома = near the house
• около десяти гостей = about ten guests

So десяти is simply the genitive form of десять.

Because after numbers like five and higher, Russian normally uses the genitive plural for the noun.

So:

• пять гостей = five guests
• десять гостей = ten guests

Here we also have около десяти, and the whole phrase stays in the structure used with quantity:

• около десяти гостей

So гостей is the correct counted form.

It can mean both, depending on context.

1. Near / next to

   • около дома = near the house

2. About / approximately

   • около десяти гостей = about ten guests

In your sentence, because около is followed by a number, it clearly means approximately.

Yes. That would also be natural.

For example:

• На празднике будет примерно десять гостей.

Both около and примерно can mean about / approximately.

A rough difference:

• около десяти гостей = around ten guests
• примерно десять гостей = approximately ten guests

Около is very common and conversational.
Примерно can sound a little more explicitly like approximately, though both are normal.

No, Russian word order is flexible.

The given sentence:

• На празднике будет около десяти гостей.

is a very natural, neutral way to say it.

Other orders are possible, for example:

• Около десяти гостей будет на празднике.
• Гостей на празднике будет около десяти.

These can shift the emphasis slightly:

• starting with На празднике highlights the setting
• starting with Около десяти гостей highlights the number
• starting with Гостей can emphasize guests as the category

But the original version is a standard, natural sentence.

It is in the prepositional case.

The dictionary form is:

• праздник

After на meaning location, it becomes:

• на празднике

So the sentence contains:

• на
  • prepositional = location
• около
  • genitive = approximation / near

This sentence is a nice example of how Russian case endings show the role of each word.

In Russian, it most naturally matches the English existential idea:

• There will be about ten guests at the party.

That is because будет here is introducing the existence of a quantity of guests at a location.

English can also say:

• About ten guests will be at the party

but that sounds a bit less neutral in many contexts. The Russian sentence is most naturally understood as the standard there will be type of sentence.