Det er min søster, der tager billeder ved stadion, mens min datter spiller, og min mor taler med en anden tilskuer.

This is a very common Danish structure called a cleft sentence.

• Min søster tager billeder ved stadion ... = a neutral statement: My sister is taking pictures by the stadium ...
• Det er min søster, der tager billeder ... = It is my sister who is taking pictures ...

The second version puts extra focus on min søster. It is like highlighting who the person is.

So Det er X, der ... often means:

• It is X who ...
• It’s X that ...

Danish uses this structure a lot, often more than English does.

In this sentence, der is the normal choice because it refers to the subject of the following clause.

In Det er min søster, der tager billeder, the word der connects back to min søster, and min søster is the one doing the action of tager.

A useful rule:

• der is commonly used when the relative word is the subject
• som can often be used in other relative clauses, especially in less fixed structures

But in the pattern Det er X, der ..., der is the standard form.

So:

• Det er min søster, der tager billeder = correct and natural

Tager billeder literally means takes pictures.

• tage = to take
• billeder = pictures/photos

So tage billeder is the normal Danish expression for to take photos.

Compare:

• tage et billede = take a picture / take one photo
• tage billeder = take pictures / take photos in general

In your sentence, the plural billeder suggests she is photographing more generally, not just taking one single photo.

Because the sentence describes the activity in a general sense.

• tage et billede = to take one picture
• tage billeder = to take pictures / do photography

So min søster tager billeder sounds like:

• she is taking photos
• she is photographing the event
• this is what she is doing right now

This is very natural in Danish.

Ved stadion means by the stadium, near the stadium, or at the stadium area.

• ved often means by, near, or at
• på stadion would more strongly suggest being at the stadium, often inside it or as part of the event location

So the difference is roughly:

• ved stadion = by/near the stadium
• på stadion = at the stadium

In real usage, both can be possible depending on context, but ved stadion gives a sense of location near the stadium rather than emphasizing being inside it.

Yes. Mens means while.

It introduces a subordinate clause:

• mens min datter spiller = while my daughter is playing

So the sentence describes two actions happening at the same time:

• the sister is taking photos
• the daughter is playing

This is one of the most common meanings of mens.

Because mens starts a subordinate clause, and Danish word order changes in subordinate clauses.

In a normal main clause, Danish usually has verb-second word order:

• Min datter spiller
• Min mor taler

But after a subordinating conjunction like mens, the order is typically:

• conjunction + subject + verb

So:

• mens min datter spiller = correct

Not:

• mens spiller min datter

A useful shortcut:

• main clauses often have verb-second order
• subordinate clauses usually do not

Actually, it already is min mor taler med en anden tilskuer—and Danish verbs do not change according to person or number the way English verbs sometimes do.

In English:

• I speak
• she speaks

In Danish:

• jeg taler
• du taler
• hun taler
• vi taler

The verb form stays the same.

So:

• min mor taler
• min søster tager
• min datter spiller

All of these are normal present tense forms, and they do not change because the subject changes.

Because tilskuer is a common-gender singular noun, so the adjective must match that form.

Danish adjective forms here are:

• en anden tilskuer = a different / another spectator
  common gender singular
• et andet sted = another place
  neuter singular
• andre tilskuere = other spectators
  plural

Since tilskuer takes en, the correct form is:

• en anden tilskuer

Tilskuer means spectator or viewer.

In this sentence, it means someone who is watching the game or event at the stadium.

Related words:

• en tilskuer = a spectator
• tilskuere = spectators
• publikum = audience / crowd / spectators as a group

So min mor taler med en anden tilskuer means she is talking to another person who is watching.

Danish uses commas more systematically than English in many cases, especially around subordinate and relative clauses.

Here the commas help separate the parts:

• Det er min søster, der tager billeder ved stadion,
• mens min datter spiller,
• og min mor taler med en anden tilskuer.

They show the boundaries between:

1. the focused opening structure
2. the relative clause with der
3. the subordinate clause with mens
4. the final coordinated clause with og

English punctuation may sometimes be lighter, but in Danish this kind of comma use is very normal.

Because the nouns are all singular and common gender:

• min søster
• min datter
• min mor

For singular nouns, Danish uses:

• min with common gender words
• mit with neuter words
• mine with plural words

Examples:

• min søster = my sister
• mit barn = my child
• mine børn = my children

So in your sentence, min is correct for all three family words.

A more neutral version would be:

Min søster tager billeder ved stadion, mens min datter spiller, og min mor taler med en anden tilskuer.

This version simply states what is happening.

The original:

• Det er min søster, der tager billeder ...

adds focus and slightly more emphasis, as if answering:

• Who is taking pictures?
• It’s my sister who is taking pictures.

So both are grammatical, but the original has a stronger focus on min søster.

Yes, functionally it works a lot like that.

The part der tager billeder ved stadion tells you more about min søster:

• she is the one taking pictures by the stadium

In English, you might think of it as:

• my sister, who is taking pictures by the stadium
• or in the cleft structure, it is my sister who is taking pictures by the stadium

So even though the whole sentence is built as a cleft, the der clause still behaves like a clause describing the person being focused on.