Die Bestellnummer steht auf dem Zahlungsbeleg.

Because Nummer is a feminine noun in German: die Nummer.

In compound nouns, the last part determines the gender.
So:

• Bestellung = order
• Nummer = number
• Bestellnummer = order number

Since the final part is Nummer, the whole word is feminine:

• die Bestellnummer

It is in the nominative case, because it is the subject of the sentence.

You can find the subject by asking: What stands/is written on the receipt?
Answer: die Bestellnummer.

That is why it stays:

• die Bestellnummer

Because auf can take either the accusative or the dative, depending on meaning.

Here it describes a location: the order number is on the receipt. There is no movement.
So German uses the dative:

• auf dem Zahlungsbeleg

Also, Zahlungsbeleg is masculine because the final noun is Beleg:

• der Beleg
• therefore: dem Zahlungsbeleg in the dative

A useful rule:

• accusative after auf = movement to a place
• dative after auf = location in/on a place

German often uses stehen for text or information that is written, printed, or displayed somewhere.

So:

• Die Bestellnummer steht auf dem Zahlungsbeleg.
  = The order number is written / appears on the receipt.

If you say:

• Die Bestellnummer ist auf dem Zahlungsbeleg.

it is understandable, but steht sounds more natural when talking about words, numbers, or information on paper or a screen.

German commonly says things like:

• Die Adresse steht auf dem Formular.
• Das Datum steht oben rechts.
• Was steht in der E-Mail?

It is a compound noun:

• Zahlung = payment
• Beleg = receipt, record, voucher, proof

So Zahlungsbeleg means something like:

• payment receipt
• proof of payment
• receipt for payment

The exact English translation can vary by context, but the German idea is basically a document showing that a payment was made.

That -s- is a common linking element in German compounds. It is often called a linking s.

So:

• Zahlung
  • Beleg → Zahlungsbeleg

You cannot always predict it perfectly from logic alone; it is just part of how many compounds are formed in German. English speakers often notice this in words like:

• Arbeitszimmer
• Lebensmittel
• Geburtsdatum

It does not usually change the meaning; it just helps form the compound.

Yes, but the finite verb must still stay in second position in a normal main clause.

Original sentence:

• Die Bestellnummer steht auf dem Zahlungsbeleg.

You could also say:

• Auf dem Zahlungsbeleg steht die Bestellnummer.

This version puts more emphasis on where it is.

What you cannot do in a normal statement is move things around and leave the verb later, like in English. German main clauses follow the verb-second rule.

Yes, auf is the natural choice here.

German uses auf for something being on a surface, including information printed on paper:

• auf dem Blatt
• auf der Quittung
• auf dem Formular

an usually suggests being attached to a vertical surface or at an edge/border, and in means in/inside.

For a number printed on a receipt, auf dem Zahlungsbeleg is the normal phrasing.

German very often forms compound nouns by writing them as one word, where English would often use two or more words.

So German writes:

• Bestellnummer
• Zahlungsbeleg

not:

• Bestell Nummer
• Zahlung Beleg

This is very common in German. A good strategy is to look at the last part first, because it usually tells you the basic category and the gender.

No. In German, all nouns are capitalized, not just the first word of the sentence.

So even in the middle of a sentence, you would write:

• Bestellnummer
• Zahlungsbeleg

This is one of the most noticeable spelling differences between German and English.

You could translate it literally that way, but it would sound unnatural in English.

German often uses stehen where English prefers expressions like:

• is on
• is written on
• appears on

So the best natural English translation is usually something like:

• The order number is on the payment receipt.
• The order number appears on the payment receipt.
• The order number is written on the payment receipt.

That is a good example of how German and English often use different verbs for the same idea.