Jeg er i tvivl om, hvilken skjorte jeg skal tage på til mødet.

Yes. At være i tvivl om is a very common fixed expression in Danish.

It means something like:

• to be unsure about
• to be uncertain about
• literally, to be in doubt about

You can use it before:

• a noun: Jeg er i tvivl om planen.
• a clause: Jeg er i tvivl om, hvilken skjorte jeg skal tage på.

So in this sentence, er i tvivl om works as one unit.

Because the expression is normally i tvivl om.

Here om means something like about or as to. It introduces the thing you are unsure about.

So:

• i tvivl om noget = unsure about something
• i tvivl om, hvad... = unsure about what...
• i tvivl om, hvilken... = unsure about which...

You normally would not leave om out here.

Because hvilken skjorte jeg skal tage på til mødet is a subordinate clause, more specifically an embedded question.

In Danish, many writers put a comma before a subordinate clause:

• Jeg er i tvivl om, hvilken skjorte jeg skal tage på til mødet.

However, Danish also allows this without the start comma:

• Jeg er i tvivl om hvilken skjorte jeg skal tage på til mødet.

So the comma is very normal, but depending on comma style, you may also see the sentence without it.

Because skjorte is a common gender noun, an en-word.

The forms are:

• hvilken for singular common gender: hvilken skjorte
• hvilket for singular neuter: hvilket bord
• hvilke for plural: hvilke skjorter

So the form of which changes to match the noun.

Because this is an embedded question, not a direct question.

In a direct question, Danish uses inversion:

• Hvilken skjorte skal jeg tage på til mødet?

But inside a larger sentence, Danish normally uses subordinate-clause word order:

• Jeg er i tvivl om, hvilken skjorte jeg skal tage på til mødet.

So after hvilken skjorte, you get:

• jeg skal...

not

• skal jeg...

This is very similar to English:

• direct: Which shirt should I put on?
• embedded: I’m not sure which shirt I should put on.

Here skal does not necessarily mean a strong must.

It often means something like:

• should
• am supposed to
• am going to
• need to decide to

In this sentence, it fits the idea of choosing what is appropriate to wear. So jeg skal tage på is very natural in context.

A good way to understand it is:

• the speaker is thinking about what shirt they are to wear for that occasion

So the nuance is practical and situational, not necessarily strict obligation.

Because tage på is the normal verb expression for put on clothes.

So:

• tage en skjorte på = put on a shirt

This is a particle verb, where på is an important part of the meaning.

Related expressions:

• tage på = put on
• have på = have on, be wearing
• klæde sig på = get dressed

So this sentence focuses on the choice of what to put on / wear.

Because tage på is a particle verb, and the particle often stays after the verb.

Basic order:

• Jeg skal tage en skjorte på.

But when the object becomes a which-phrase, it moves to the front of the embedded clause:

• hvilken skjorte jeg skal tage på

So the shirt is mentioned first, but på still stays with the verb phrase at the end.

This is actually similar to English:

• which shirt I should put on

Because til mødet means for the meeting or to the meeting, in the sense of the occasion the clothes are for.

That is exactly what you want here: the speaker is choosing a shirt for that event.

Compare:

• til mødet = for the meeting
• på mødet = at the meeting

So:

• Jeg er i tvivl om, hvilken skjorte jeg skal tage på til mødet.
  = I’m unsure which shirt to wear for the meeting.

But:

• Jeg havde en blå skjorte på på mødet.
  = I was wearing a blue shirt at the meeting.

Because the sentence refers to a specific meeting.

In Danish, the definite form is often used when the speaker has a particular thing in mind:

• mødet = the meeting

If it were a non-specific meeting, you could say:

• til et møde = for a meeting

So the choice between mødet and et møde depends on whether the meeting is specific and known in context.

Because hvilken already functions as the determiner.

Just as English says:

• which shirt

Danish says:

• hvilken skjorte

You do not add en here, and you do not use the definite ending either.

So these are not correct in this structure:

• hvilken en skjorte
• hvilken skjorten

The normal form is simply:

• hvilken skjorte