Min læge siger, at en måned går hurtigt, når man har meget at se til.

Because læge is a common-gender noun in Danish: en læge = a doctor. With common-gender singular nouns, you use min for my.

So:

• en læge → min læge
• et hus → mit hus

Also, Danish does not use an article after a possessive, so min læge is correct, not min en læge.

Here at means that and introduces a subordinate clause:

Min læge siger, at ...
= My doctor says that ...

English often leaves out that, especially in speech, but Danish commonly keeps at in this kind of sentence.

En måned means a month, so it is being used in a general sense: a month goes by quickly.

If you said måneden, that would mean the month, usually a specific month already known from the context.

So:

• en måned = a month, any month / a month in general
• måneden = the month, a specific one

Yes, and that is a very natural Danish way to talk about time passing. Danish often uses gå with periods of time:

• Tiden går hurtigt = Time passes quickly
• Dagen går langsomt = The day passes slowly
• En måned går hurtigt = A month goes quickly / passes quickly

So although it is literal in Danish, the natural English translation is usually passes quickly.

Because hurtigt is functioning as an adverb here: it describes how the month goes/passes.

• hurtig = quick / fast as an adjective
• hurtigt = quickly / fast as an adverb

In this sentence, hurtigt modifies the verb går, not the noun måned.

Compare:

• en hurtig måned = a quick month
• en måned går hurtigt = a month passes quickly

Here når means when.

It is used because the sentence expresses a general truth or repeated situation: a month goes quickly when people are very busy.

Hvis means if, and it would sound different:

• når = when, in situations that happen or generally apply
• hvis = if, under a condition

So in this sentence, når is the natural choice.

No. In Danish, man is often an impersonal pronoun meaning one, you, or people in general.

So:

• når man har meget at se til
  means something like
• when one has a lot to do
• when you have a lot on
• when people are very busy

It does not specifically mean a male person here.

It is an idiomatic expression. At have meget at se til means to have a lot to do, to be very busy, or to have a lot to attend to.

It is not meant literally as to have much to look at.

The expression se til can have meanings like look after, attend to, or deal with, and in this fixed phrase it gives the idea of being busy with many responsibilities.

A close alternative would be:

• man har meget at lave = one has a lot to do

Because når introduces a subordinate clause, and Danish word order changes in subordinate clauses.

In a main clause, Danish usually has verb second:

• Min læge siger ...

But in a subordinate clause introduced by words like at or når, the normal order is:

• conjunction + subject + verb

So:

• at en måned går hurtigt
• når man har meget at se til

That is why you get når man har, not når har man.

Because both at en måned går hurtigt and når man har meget at se til are subordinate clauses, and many Danish writers place commas before subordinate clauses.

You will see both styles in Danish:

• Min læge siger, at ...
• Min læge siger at ...

Likewise:

• ..., når man har ...
• ... når man har ...

So the commas in your sentence are normal and correct. Danish comma usage can vary depending on the comma system being followed.

Danish present tense often covers both ideas, depending on context.

So siger can correspond to:

• says
• is saying

In this sentence, says is the most natural English choice, because it sounds like a general statement: My doctor says that ...

Yes. A common simpler alternative is man er travl, which means one is busy.

So you could say:

• Min læge siger, at en måned går hurtigt, når man er travl.

That is grammatical and natural, but the nuance is slightly different:

• har meget at se til focuses on having many things to deal with
• er travl simply states that someone is busy

It breaks down like this:

• Min læge siger = main clause
• at en måned går hurtigt = subordinate clause after siger
• når man har meget at se til = subordinate clause explaining when that happens

So the sentence is basically:

My doctor says

• that a month passes quickly
• when one has a lot to do

This is a very common Danish pattern: main clause + at-clause + optional når-clause.