Open je mond voor de tandarts.

Open is the imperative form, the form used for commands and instructions.

• openen = to open
• open = open!

So Open je mond is the natural way to tell someone to open their mouth.

opent is not the normal informal command form. It is a present-tense form used with u or with hij/zij/het in regular statements.

In Dutch, just like in English, commands often leave out the subject.

• English: Open your mouth.
• Dutch: Open je mond.

The understood subject is you, but it does not need to be said.

Here je means your.

Dutch je can be both:

• a subject/object pronoun: you
• a possessive determiner: your

In Open je mond, it is possessive, so it belongs with mond:

• je mond = your mouth

Je is the normal, unstressed way to say your.
Jouw is usually used when you want extra emphasis or contrast.

So:

• Open je mond. = normal, neutral
• Open jouw mond. = more emphatic, like open your mouth

A dentist would normally say je mond, not jouw mond.

Because a possessive word like je replaces the article.

You say:

• de mond = the mouth
• je mond = your mouth

But not:

• de je mond ✗

So once you use je, you do not also use de.

It is a de word:

• de mond

In this sentence, you do not see de because the possessive je replaces it.

Here voor means for in the sense of for the dentist / so the dentist can examine you.

So Open je mond voor de tandarts means something like:

• open your mouth for the dentist
• open your mouth so the dentist can look

It is not mainly about physical position here. It is about the purpose or the person the action is for.

Because tandarts is a de noun:

• de tandarts = the dentist

Also, tandarts can refer to either a male or female dentist. The article de does not mean the dentist is male.

Yes. That is also very common.

• Open je mond.
• Doe je mond open.

Both are natural Dutch.
Doe je mond open can sound a little more conversational in everyday speech, and you may also hear:

• Doe je mond maar open.

A dentist might use either version.

A polite version would be:

• Opent u uw mond.

You may also hear more polite or softer versions such as:

• Wilt u uw mond openen?
• Doet u uw mond open.

Here:

• u = polite you
• uw = polite your

So je mond becomes uw mond in formal speech.