Wir beschriften den Fotoabzug mit dem Aufnahmedatum.

Because Fotoabzug is the direct object of beschriften, so it takes the accusative case.

• der Fotoabzug = nominative singular
• den Fotoabzug = accusative singular

The verb beschriften means something like to label / to write on / to mark with writing, and the thing being labeled is the object of the verb.

So:

• Der Fotoabzug ist neu. = The photo print is new.
• Wir beschriften den Fotoabzug. = We label the photo print.

Since Fotoabzug is masculine (der), its article changes from der to den in the accusative.

Because the preposition mit always takes the dative case.

So:

• das Aufnahmedatum = nominative
• dem Aufnahmedatum = dative

That is why you get:

• mit dem Aufnahmedatum = with the date the photo was taken

This is a very common pattern in German:

• mit dem Stift = with the pen
• mit der Hand = with the hand
• mit dem Namen = with the name

Here, mit tells you what information is used to label the print.

Beschriften means to label, to mark with writing, or to write identifying information on something.

It is built from:

• Schrift = writing / script
• the verb prefix be-, which often turns something into a verb that acts directly on an object

So etwas beschriften means to put writing on something in order to identify it.

In this sentence, it does not just mean random writing. It suggests adding a useful label or identifying text.

For example:

• Wir beschriften die Kisten. = We label the boxes.
• Sie beschriftet das Heft mit ihrem Namen. = She labels the notebook with her name.

A Fotoabzug is a photo print, especially a print made from a photograph.

It is more specific than just Foto:

• das Foto = the photo / photograph
• der Fotoabzug = the physical printed copy of the photo

So the sentence is talking about labeling the printed photo, not necessarily the image in general.

You may also hear:

• der Abzug = print
• das Bild = picture/image
• das Foto = photo

But Fotoabzug is a good, precise word for a printed photographic copy.

Because German very often makes compound nouns by joining words together.

Here:

• die Aufnahme = recording, shot, photo taken
• das Datum = date

Together:

• das Aufnahmedatum = the date of the shot / the date the photo was taken

English usually uses separate words or an of phrase, but German prefers compounds.

Other examples:

• Geburtsdatum = date of birth
• Straßenname = street name
• Hausnummer = house number

When the final part of the compound is Datum, the whole noun is neuter, because the last element determines the gender:

• das Datum
• therefore das Aufnahmedatum

Yes. Aufnahme has several meanings depending on context, including:

• recording
• shot
• photograph taken
• admission/intake in some contexts

In this sentence, because of Fotoabzug, it clearly refers to the taking of the photo. So Aufnahmedatum means the date on which the photo was taken, not the date it was printed.

Yes. das Datum der Aufnahme is grammatical and means essentially the same thing.

So these are both possible:

• mit dem Aufnahmedatum
• mit dem Datum der Aufnahme

The compound Aufnahmedatum sounds compact and very natural, especially in written or technical contexts. German often prefers the compound form where English might use an of phrase.

It is present tense:

• wir beschriften = we label / we are labeling

Like English, German present tense can sometimes correspond to more than one English form depending on context.

So the sentence could mean:

• We label the photo print with the date it was taken.
• We are labeling the photo print with the date it was taken.

German does not need a separate progressive form like English are labeling.

Yes. German word order is flexible, as long as the finite verb stays in the correct position in a main clause.

The original sentence is:

• Wir beschriften den Fotoabzug mit dem Aufnahmedatum.

You could also say:

• Den Fotoabzug beschriften wir mit dem Aufnahmedatum.
• Mit dem Aufnahmedatum beschriften wir den Fotoabzug.

These versions change the emphasis:

• starting with den Fotoabzug puts more focus on the photo print
• starting with mit dem Aufnahmedatum puts more focus on the information used for labeling

But in all of them, beschriften stays in the second position as the finite verb.

More or less, yes, but it is worth noticing the exact sense.

Here mit introduces the thing used in the labeling:

• mit dem Aufnahmedatum = using the date of the shot as the label/information

So it is not necessarily a physical tool, as in:

• mit einem Stift = with a pen

It can also introduce the content or identifying information:

• mit seinem Namen = with his name
• mit einer Nummer = with a number

That is exactly how it works in this sentence.

Normally, in a full standard sentence like this, you would say mit dem Aufnahmedatum.

Leaving out the article would sound incomplete or unnatural in ordinary prose. However, article-less forms can appear in:

• headings
• labels
• notes
• very compressed technical style

For a normal sentence, mit dem Aufnahmedatum is the expected form.

Because Aufnahmedatum is a neuter singular noun, and in modern German the noun itself usually does not change in the dative singular. The article shows the case:

• das Aufnahmedatum = nominative/accusative singular
• dem Aufnahmedatum = dative singular

So the important visible change is mainly:

• das → dem

The noun stays the same.