A teljes zenekar a színpadon van.

Because Hungarian uses the definite article before each definite noun phrase.

In this sentence:

• a teljes zenekar = the whole orchestra
• a színpadon = on the stage / on stage

So the two a words are not duplicates by mistake; each belongs to a different part of the sentence.

Also, the Hungarian definite article has two forms:

• a before a consonant
• az before a vowel

So here it is a teljes, a színpadon, because teljes and színpadon both begin with consonants.

Teljes here means whole, entire, or full.

So a teljes zenekar means the entire orchestra.

Yes, az egész zenekar is also possible and very natural. The difference is small in many contexts, but there can be a nuance:

• egész often emphasizes the whole thing as one unit
• teljes often emphasizes completeness or fullness

In this sentence, both can work, but teljes may sound a little more formal or slightly more emphatic.

Because zenekar is a singular collective noun, like orchestra in English.

Hungarian treats a teljes zenekar as one unit, so the verb is singular too:

• A teljes zenekar ... van.

If you wanted to talk about the individual people, you would use a plural noun instead, for example a zenészek = the musicians.

Yes. It is:

• színpad = stage
• -on = a location ending meaning on

So:

• színpadon = on the stage

This ending belongs to the Hungarian case system. The basic pattern is:

• -on / -en / -ön

This case is called the superessive, and it often corresponds to English on.

Because Hungarian uses different endings depending on the kind of location.

• -ban / -ben = in, inside something
• -on / -en / -ön = on, on the surface of something

A stage is normally thought of as a surface or platform, so Hungarian says:

• a színpadon = on the stage

Using színpadban would suggest being physically inside the stage, which is not the normal meaning here.

That is true in some sentences, but not in this kind.

Hungarian often omits van in the present tense, third person, when the sentence simply identifies or describes something:

• A zenekar nagy. = The orchestra is big.
• A karmester fáradt. = The conductor is tired.

But when the sentence says where something is, van is normally kept:

• A zenekar a színpadon van. = The orchestra is on stage.

So here van is necessary because this is a location sentence.

Hungarian word order is flexible, but changing it changes the emphasis.

The sentence as given:

• A teljes zenekar a színpadon van.

is a neutral way to say it.

Other orders are possible, for example:

• A színpadon van a teljes zenekar.
  This can emphasize the location, as if answering Where is the whole orchestra?

• A teljes zenekar van a színpadon.
  This can emphasize the whole orchestra, perhaps in contrast to only part of it.

So the words can move, but the version you were given is a normal, neutral sentence.

Because article usage does not match exactly between English and Hungarian.

Hungarian often uses the definite article in places where English may leave it out in an idiomatic expression. So:

• a színpadon van

can correspond to either:

• is on the stage
• is on stage

depending on the English style and context.

So even if English drops the article, Hungarian may still keep a.