Tandlægen skal bore i tanden.

Because -en is the usual definite ending for many common-gender Danish nouns.
So:

• en tandlæge = a dentist
• tandlægen = the dentist

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

It is a compound noun:

• tand = tooth
• læge = doctor

So tandlæge literally means something like tooth doctor, which is the Danish word for dentist.

In Danish, you normally do not use en/et together with the simple definite form.

So:

• en tand = a tooth
• tanden = the tooth

You would not normally say en tanden.

The ending -en already carries the idea of the.

Skal is the present tense of skulle. In this sentence, it means something like:

• must
• has to
• is supposed to
• sometimes is going to

So Tandlægen skal bore i tanden suggests that the dentist is expected or required to do the drilling.

Because after a modal verb like skal, Danish uses the infinitive form of the next verb.

So:

• skal bore = must drill / is going to drill

Compare:

• Tandlægen borer i tanden = The dentist is drilling / drills in the tooth
• Tandlægen skal bore i tanden = The dentist must drill / is going to drill in the tooth

After skal, you do not use the present tense borer.

Because after modal verbs such as skal, kan, vil, må, and bør, Danish normally uses the bare infinitive.

So you say:

• skal bore
• kan bore
• vil bore

not:

• skal at bore

This works a bit like English must drill, not must to drill.

Because bore i means to drill into / in something.

With dental work, Danish commonly uses i tanden because the drilling is understood as going into the tooth, not just happening on its surface.

So:

• bore i tanden = drill into the tooth

Using på would sound less natural here.

Yes, but only in the drilling sense.

Danish bore means to drill. It is related to English bore as in to bore a hole.

It does not mean to make someone feel bored.

The basic order here is:

• Tandlægen = subject
• skal = finite verb
• bore = infinitive/main verb
• i tanden = prepositional phrase

So the pattern is:

Subject + finite verb + infinitive + other information

This is a normal Danish main-clause pattern.

In a yes/no question, Danish usually puts the finite verb before the subject:

• Skal tandlægen bore i tanden? = Is the dentist going to drill in the tooth? / Must the dentist drill in the tooth?

This verb-before-subject pattern is very common in Danish questions.

You usually put ikke after the finite verb and before the infinitive/main content:

• Tandlægen skal ikke bore i tanden.

That means:

• The dentist must not drill in the tooth
• or The dentist is not going to drill in the tooth

The exact meaning depends on context.

A rough learner-friendly guide would be:

• tand ≈ tan(d), with the d not strongly pronounced like English d
• læ has the Danish æ, a vowel somewhat like the a in cat, but not exactly
• gen is softer than an English learner may expect

Very roughly: TAN-lai-yen is not correct, and English spelling cannot show it well.
A better rough idea is something like TAN-læ-yen, with a distinctly Danish æ sound.

The hardest parts for English speakers are usually:

• the vowel æ
• the softer Danish consonants
• keeping the word smooth rather than heavily stressed syllable by syllable

A rough guide is BOH-rə.

Important points:

• the first syllable has a long o-sound
• the final -e is not silent; it is a light schwa-like sound

So it is closer to BOH-ruh than to English bore.

Sometimes Danish prefers the preposition because the action is understood as drilling into the tooth:

• bore i tanden

Without i, the sentence may sound less idiomatic in this context.
So for a learner, bore i tanden is the safer and more natural choice when talking about dental drilling.

Because the sentence is talking about specific, identifiable things:

• tandlægen = the dentist
• tanden = the tooth

In context, both are understood as known or specific. Danish often uses the definite form this way when the speaker has a particular person and object in mind.