Jag har alltid en näsduk i väskan när det blåser kallt ute.

Because Swedish usually puts the finite verb in the second position in a main clause. This is often called the V2 rule.

So in:

Jag har alltid en näsduk ...

• Jag = subject
• har = finite verb
• alltid = adverb

That order is very normal in Swedish.

Compare:

• Jag har ofta tid.
• Hon kommer aldrig sent.

If you move another part to the front, the verb still stays second:

• När det blåser kallt ute har jag alltid en näsduk i väskan.

So alltid after har is the expected word order.

Because näsduk is a countable singular noun, and in Swedish you usually need an article in that situation.

• en näsduk = a handkerchief / a tissue
• näsduken = the handkerchief / the tissue

You normally cannot say just jag har alltid näsduk here.

Also, näsduk is an en-word, so the indefinite singular form is en näsduk, not ett näsduk.

I väskan literally means in the bag. In context, it can easily mean in my bag if it is obvious whose bag is meant.

Swedish sometimes uses the definite form where English would prefer a possessive, especially when the context makes the owner clear.

So:

• i väskan = in the bag
• i min väska = in my bag

Both can work, but i väskan can sound natural if the speaker’s bag is already understood.
If you want to be more explicit, i min väska is also perfectly good Swedish.

Because Swedish often uses det as a dummy subject in weather expressions, just like English uses it in it is raining or it is windy.

So:

• det regnar = it is raining
• det snöar = it is snowing
• det blåser = it is blowing / it is windy

The det does not refer to a specific thing. It is just there because Swedish sentences normally need a subject.

Because kallt is the form used adverbially here.

The basic adjective is:

• kall = cold

But when Swedish uses it to describe how something feels or happens, the -t form is often used:

• Det är kallt.
• Det blåser kallt.

In det blåser kallt, the idea is roughly the wind is blowing cold or it is blowing cold.

So kallt is not agreeing with a noun here. It is describing the situation more generally, which is why the -t form appears.

Ute means outside / outdoors.

In weather expressions, Swedish very often adds ute to make it clear that the weather is being described outside:

• Det är kallt ute. = It is cold outside.
• Det blåser ute. = It is windy outside.

So in:

när det blåser kallt ute

ute gives the natural idea of outdoor conditions. Without it, the sentence would still be understandable, but ute makes it sound more complete and idiomatic.

Because när here means when / whenever, referring to a situation that really happens, especially a repeated one.

The sentence means something like:

I always have a handkerchief in my bag whenever it is blowing cold outside.

Use när for repeated or expected situations:

• När jag är trött, dricker jag kaffe.
• När det regnar, stannar vi inne.

Use om more for if in a conditional sense:

• Om det regnar i morgon, stannar vi inne.

So när is the natural choice for a habitual pattern.

Not exactly.

• det är kallt ute = it is cold outside
• det blåser kallt ute = a cold wind is blowing outside / it is blowing cold outside

The second one focuses more on the wind and the feeling of cold air blowing.
So it is more specific than just saying that the temperature outside is low.

Yes. Swedish word order can change depending on what you want to emphasize, but the verb-position rules still apply.

For example, you can say:

När det blåser kallt ute har jag alltid en näsduk i väskan.

This puts the time/condition part first: when it is blowing cold outside.

Notice what happens:

• the subordinate clause När det blåser kallt ute comes first
• the finite verb in the main clause, har, still comes before the subject jag

That is a very typical Swedish pattern.

It can refer to either, depending on context, though many learners first meet it as handkerchief.

In modern everyday use, if you specifically mean a paper tissue, people may also say:

• pappersnäsduk = paper tissue

But näsduk is still a normal general word and works fine in a sentence like this.

So the exact image can depend on context, but grammatically the sentence works the same way.