I tandlægestolen undersøger tandlægen tanden og tager et røntgenbillede.

This is because Danish main clauses normally follow the V2 rule: the finite verb must come in second position.

• Neutral order: Tandlægen undersøger tanden.
• If you move another element to the front, such as a place phrase: I tandlægestolen undersøger tandlægen tanden.

So when I tandlægestolen is placed first, the verb undersøger must still stay second, and the subject tandlægen moves after it.

This is very common in Danish and is one of the biggest word-order differences from English.

The ending -en is the definite singular article for many Danish nouns. It means the.

So:

• tandlægestol = dentist’s chair / dental chair

• tandlægestolen = the dentist’s chair / the dental chair

• tandlæge = dentist

• tandlægen = the dentist

• tand = tooth
• tanden = the tooth

Unlike English, Danish often puts the at the end of the noun instead of as a separate word before it.

Because in Danish, the definite article is usually attached to the noun.

So English:

• the dentist
• the tooth

becomes Danish:

• tandlægen
• tanden

You normally do not say den tandlægen or den tanden in a plain sentence.

A separate article such as den is used when there is an adjective:

• den store tandlægestol = the big dental chair

So:

• without adjective: tandlægestolen
• with adjective: den store tandlægestol

Because røntgenbillede is a neuter noun, so it takes et.

Danish nouns have grammatical gender, and singular indefinite articles are:

• en for common gender
• et for neuter

Here, the important part is billede = picture/image, which is neuter:

• et billede
• therefore et røntgenbillede

In compounds, the last part usually determines the gender. Since billede is neuter, røntgenbillede is also neuter.

Because Danish forms compound nouns by writing them as one word.

So:

• tandlæge
  • stol → tandlægestol
• røntgen
  • billede → røntgenbillede

This is very normal in Danish. English often writes similar expressions as two words or with an apostrophe, but Danish usually joins them.

The last part is the main noun:

• tandlægestol is a kind of stol
• røntgenbillede is a kind of billede

Then the definite ending can be added:

• tandlægestolen = the dental chair

Here i means in.

With chairs and seats, Danish often uses i when you mean sitting down in something that surrounds or holds you, especially with larger or more enclosing seats:

• i stolen = in the chair
• i sofaen = on/in the sofa
• i tandlægestolen = in the dentist’s chair

Using på would suggest more literally being on top of something. So i tandlægestolen sounds natural here.

They are both present tense forms.

• at undersøge = to examine

• undersøger = examines / is examining

• at tage = to take
• tager = takes / is taking

In Danish, the present tense is often formed with -r, and the verb does not change for person:

• jeg tager
• du tager
• tandlægen tager

This is simpler than English, because Danish does not have different present-tense forms like I take vs he takes.

Because the same subject is understood for both verbs.

So:

• tandlægen undersøger tanden og tager et røntgenbillede

means:

• the dentist examines the tooth and takes an X-ray

Just like in English, you do not need to repeat the subject when one subject does two actions joined by og = and.

You could repeat it, but it would usually sound less natural unless you wanted special emphasis.

Because they play different roles in the sentence.

• tanden = the tooth
  This refers to a specific tooth being examined, so the definite form is natural.

• et røntgenbillede = an X-ray
  This introduces one X-ray as an action/result, so the indefinite form is natural.

If you said røntgenbilledet, that would mean the X-ray, as if the listener already knew which X-ray you meant.

So the contrast is very normal:

• known/specific thing: tanden
• newly introduced item: et røntgenbillede

Yes, that is grammatically possible, but it changes the emphasis a little.

• I tandlægestolen undersøger tandlægen tanden ...
  This puts the location/setting first. It sounds like In the dentist’s chair, ...

• Tandlægen undersøger tanden i tandlægestolen ...
  This starts more neutrally with the subject: The dentist examines the tooth ...

So both can work, but the original sentence highlights the place first, while the alternative sounds more straightforward and neutral.

Yes, it is a very common and natural expression.

Literally, tage means take, and Danish often uses it in the same way English does in medical or technical contexts:

• tage et billede = take a picture
• tage et røntgenbillede = take an X-ray

So even if the equipment is producing the image, Danish still normally says tage here. It is the standard verb to use.