Uanset om hun er træt eller har travlt, vinker hun altid til naboen om morgenen.

Because Danish main clauses normally follow the verb-second rule.

In this sentence, the opening part Uanset om hun er træt eller har travlt takes the first position in the sentence. That means the finite verb of the main clause must come next:

• Uanset om ..., vinker hun altid ...

So the order is:

1. fronted clause
2. finite verb: vinker
3. subject: hun

If you remove the opening clause, you get the more basic order:

• Hun vinker altid til naboen om morgenen.

Both are normal; the first one just has inversion because something else comes first.

Uanset om means regardless of whether or whether or not.

It introduces two possible situations, and the main clause is true in both cases:

• hun er træt = she is tired
• har travlt = is busy / is in a hurry

So Uanset om hun er træt eller har travlt means that it does not matter which of those is true: she still waves.

This is a very common pattern in Danish:

• Uanset om det regner eller sner, går han en tur.
  = Regardless of whether it rains or snows, he goes for a walk.

Because Danish uses two different constructions here.

• at være træt = to be tired
• at have travlt = to be busy / to be in a hurry

So:

• hun er træt = she is tired
• hun har travlt = she is busy

Even though English often uses be in both cases, Danish does not. Have travlt is a fixed expression, so you simply have to learn it as a unit.

In the expression have travlt, Danish uses travlt, not travl.

This is a fixed idiomatic phrase:

• jeg har travlt
• hun har travlt
• de har travlt

It does not change to match the subject here. So even though the basic adjective is travl (busy), the expression you learn is at have travlt.

Because the same subject applies to both parts.

In hun er træt eller har travlt, the subject hun belongs to both verbs:

• hun er træt
• (hun) har travlt

This works just like English:

• whether she is tired or busy

You could repeat the subject, but it would usually sound unnecessary here.

Because the Danish verb at vinke normally takes the preposition til when you wave to someone.

So:

• vinke til nogen = wave to someone

Examples:

• Hun vinkede til mig. = She waved to me.
• Børnene vinker til læreren. = The children wave to the teacher.

So til naboen is not optional here; it is the normal construction with this meaning.

Naboen means the neighbor.

The base noun is:

• en nabo = a neighbor

The ending -en makes it definite:

• naboen = the neighbor

This is very common in Danish. Instead of a separate word like English the, Danish often adds the definite ending to the noun:

• en bil → bilen
• en nabo → naboen

Om morgenen means in the morning or, in a sentence like this, often in the mornings / every morning.

This is a common Danish time expression:

• om morgenen = in the morning
• om aftenen = in the evening
• om natten = at night

Here it describes a habitual action, especially because the sentence also has altid:

• vinker hun altid ... om morgenen
  = she always waves ... in the morning

A useful contrast:

• i morgen = tomorrow
• om morgenen = in the morning

Those look similar, so learners often mix them up.

Altid usually comes after the subject in a main clause.

So in a normal sentence:

• Hun vinker altid til naboen.

But if another element comes first and causes inversion, the verb comes before the subject, and altid still comes after the subject:

• Uanset om ..., vinker hun altid til naboen.

So here the order is:

• vinker = finite verb
• hun = subject
• altid = adverb

That placement is very natural in Danish.

Because the first part is a subordinate clause placed before the main clause.

The structure is:

• subordinate clause: Uanset om hun er træt eller har travlt
• main clause: vinker hun altid til naboen om morgenen

The comma marks the boundary between them.

So the comma is helping show:

• Regardless of whether she is tired or busy,
• she always waves to the neighbor in the morning.

No. Danish om can have different meanings depending on the construction.

In uanset om, it means whether, not about.

Compare:

• Vi taler om bogen. = We are talking about the book.
• Jeg ved ikke, om hun kommer. = I don't know whether she is coming.

So in this sentence, om is part of the set phrase uanset om and means whether.