Заказ песни уже принят.

Because the noun заказ usually takes a dependent noun in the genitive:

• заказ песни = an order/request for a song
• заказ билета = a ticket order
• заказ еды = a food order

So:

• заказать песню = to order/request a song
  
  • here песню is the direct object, so it is accusative
• заказ песни = the order/request of a song
  
  • here песни depends on the noun заказ, so it is genitive

This is one of the most common noun + genitive patterns in Russian.

Принят is the short passive participle of принять.

It means something like:

• accepted
• received
• taken

in the sense of an order/request being officially handled.

So Заказ песни уже принят literally has the structure:

• The song order/request is already accepted

but in natural English it is usually rendered as:

• The song request has already been accepted
• Your song order has already been taken

Because принял and принят are different forms:

• принял = accepted / took
  This is a past tense active verb. It means someone did the action.
  Example: Администратор принял заказ. = The administrator accepted the order.

• принят = accepted
  This is a short passive participle. It describes the order as being in the accepted state.
  Example: Заказ принят. = The order has been accepted.

In your sentence, the focus is on the order, not on who accepted it, so Russian uses принят.

In Russian, short passive participles are often used without быть when the idea is a current result or status.

So:

• Заказ принят. = The order has been accepted. / The order is accepted.
• Заказ уже принят. = The order has already been accepted.

If you say был принят, that is a more clearly marked past form:

• Заказ был принят вчера. = The order was accepted yesterday.

So in your sentence, leaving out был is completely normal and natural.

Yes. Принят agrees with заказ in gender and number.

Here:

• заказ is masculine singular
• so the short participle is принят

Compare:

• заявка принята = the request/application has been accepted
  (заявка is feminine)
• письмо принято = the letter has been accepted
  (письмо is neuter)
• заказы приняты = the orders have been accepted
  (plural)

So the ending of the short participle changes to match the noun.

Уже means already.

It shows that the action/result happened before now, often earlier than expected or earlier than the listener might think.

So:

• Заказ песни принят. = The song request has been accepted.
• Заказ песни уже принят. = The song request has already been accepted.

It often gives a slight sense of:

• it’s done now
• this step is completed
• you don’t need to worry about it

Russian word order is flexible, but the version you have is a very neutral, natural one:

• Заказ песни уже принят.

Other orders are possible, but they change the emphasis:

• Уже принят заказ песни.
  More emphasis on the fact that it is already accepted

• Заказ песни принят уже.
  Less neutral; possible in some contexts, but not the most standard choice here

So the given order is a good default sentence.

The participle принят comes from the perfective verb принять.

That means the sentence presents the action as completed and focuses on the result:

• the order/request is now in the state of being accepted

That is why English often translates it with:

• has been accepted

rather than something ongoing like:

• is being accepted

If Russian wanted an ongoing/process meaning, it would use a different form, for example:

• Заказ песни принимается. = The song order/request is being accepted.

So принят is about the finished result.

Yes, it can sound natural, especially in contexts where someone requests or orders a song, for example:

• on the radio
• in karaoke
• at a live event
• in a music request system

Depending on the exact situation, Russian might also use other nouns, such as:

• заявка на песню = a request for a song
• заказ песни = an order/request of a song

But заказ песни уже принят is a perfectly understandable and natural phrase in the right context.

If you want to say who did the action, Russian can use an active construction instead of the passive one:

• Мы уже приняли заказ песни. = We have already accepted the song request.
• Администратор уже принял заказ песни. = The administrator has already accepted the song request.

A very common Russian alternative is also:

• Заказ песни уже приняли.

Literally this is They already accepted the song request, but in Russian it often works like an indefinite-personal sentence, where the doer is not specified. In English, it can still be translated passively:

• The song request has already been accepted.

A natural stressed version is:

Зака́з пе́сни уже́ при́нят.

Word stress:

• зака́з
• пе́сни
• уже́
• при́нят

This can help because Russian stress is not always predictable, and learners often want to know where the emphasis falls.