Learn Hardcore Swedish: På vintern tar jag alltid mössa och halsduk när jag går den där vägen. - In the winter I always take a hat and a scarf when I walk that road.

i vintern is not idiomatic in this meaning. You can say under vintern (“during the winter”) if you want to emphasize the whole period, but på vintern is the most natural for general habits.

Why is it tar jag alltid and not jag tar alltid after på vintern?

Swedish has V2 word order in main clauses: the finite verb must be in second position, no matter what comes first.

Neutral order: Jag tar alltid mössa…
(subject first, verb second – fine)
With an adverbial (time expression) first:
På vintern tar jag alltid mössa…
(1st: På vintern, 2nd: tar, 3rd: jag…)

So when you move På vintern to the front, the verb tar has to move to the second position, before jag.

På vintern jag tar alltid… is ungrammatical in Swedish.

What exactly does tar mean here? Is it “take”, “wear”, or “bring”?

Literally, ta means “to take”, but depending on context it can overlap with English bring or wear.

In this sentence:

På vintern tar jag alltid mössa och halsduk…

the idea is:

I always bring / put on / wear a hat and scarf
(because it’s cold when I go that way)

Very common expressions are:

Jag tar med mig en jacka. – I’m bringing a jacket (with me).
Jag tar på mig en jacka. – I put on a jacket.

Your sentence omits på mig or med mig, because it’s obvious from context that it’s about wearing/bringing them. In everyday speech this is quite natural: Jag tar mössa och halsduk.

Why is there no article: mössa and halsduk, not en mössa and en halsduk?

With clothes in Swedish, when you talk about what you wear or take (for yourself), you often drop both:

the indefinite article (en/ett) and
the possessive (min/mitt/mina)

So instead of saying:

Jag tar en mössa och en halsduk.
Jag tar min mössa och min halsduk.

you can simply say:

Jag tar mössa och halsduk.

This is very natural and sounds generic: “I take (some) hat and scarf for myself.”

Compare:

Jag har mössa. – I’m wearing a hat.
Jag har en mössa. – I have a hat (I own one).

The version without the article is often about the current outfit, not ownership.

Could I say min mössa och min halsduk instead? What’s the difference?

Yes, you can say:

På vintern tar jag alltid min mössa och min halsduk…

That emphasizes that it is specifically your own particular hat and scarf, not just any hat and scarf.

Nuance:

mössa och halsduk (no possessive) – typical, general clothing you put on yourself; the “normal” way to say it.
min mössa och min halsduk – more specific; maybe you are talking about those particular items (e.g., the red hat and the blue scarf you always use).

Why is it när jag går den där vägen and not när jag går på den där vägen?

In Swedish, gå can take a direct object when it means “to go / walk a certain road/route”:

gå vägen – go/walk that way, take that road/route
gå den här vägen – go this way
gå den där vägen – go that way (over there / that specific route)

So:

när jag går den där vägen
literally: when I walk that road/that route

You can say gå på vägen sometimes, but that focuses more on moving on the surface of the road (as a physical place). In this sentence, den där vägen is more like “that route I sometimes take” – so no preposition is used.

What is the nuance of den där vägen compared to den vägen or den här vägen?

Demonstratives in Swedish:

den här vägen – this road/way (near me / just mentioned)
den där vägen – that road/way (farther away / more “over there”)
den vägen – that/that particular road (neutral, just specific)

den där vägen often suggests:

the speaker and listener both know which road is meant, and
it is somewhat distanced (farther away, or emotionally/mentally marked as “that one there”).

So:

när jag går den där vägen = when I go that particular way (you know the one I mean).

Why is the word order när jag går den där vägen, not när går jag den där vägen?

After när (when) introducing a subordinate clause, Swedish uses subject–verb order (no V2 rule there):

när jag går den där vägen
(jag = subject, går = verb)

If när is a question word, then you do get V2:

När går jag den där vägen? – When do I walk that way? (a question)

So in your sentence när is not asking a question; it’s introducing a time clause:

På vintern tar jag alltid mössa och halsduk när jag går den där vägen.
= In winter, I always take a hat and scarf when I walk that way.

Could I use åker instead of går here? What’s the difference?

gå = go on foot, walk
åka = go by vehicle (car, bus, train, bike, etc.)

So:

när jag går den där vägen – when I walk that way/that road
när jag åker den där vägen – when I travel (by vehicle) that way/that road

Because the sentence talks about a mössa och halsduk (hat and scarf), it sounds natural that the person is walking in the cold. åker would be possible but gives a slightly different image: taking that route by car/bus etc.

Could I say På vintern brukar jag ta mössa och halsduk instead of tar jag alltid?

You can, but the meaning changes slightly:

tar jag alltid – I always take (100% of the time, very strong)
brukar jag ta – I usually / generally take (a habit, but not necessarily every single time)

So:

På vintern tar jag alltid mössa och halsduk när jag går den där vägen.
= In winter I always take a hat and scarf when I walk that way.
På vintern brukar jag ta mössa och halsduk när jag går den där vägen.
= In winter I usually take a hat and scarf when I walk that way.

Both are grammatically fine; it depends how strong you want to state the habit.

If I want to stress that I wear them, not just carry them, how could I say that in Swedish?

To make it very clear that you put them on your body, you can use ta på mig or ha på mig:

På vintern tar jag alltid på mig mössa och halsduk när jag går den där vägen.
= In winter I always put on a hat and scarf when I walk that way.
På vintern har jag alltid mössa och halsduk på mig när jag går den där vägen.
= In winter I always wear a hat and scarf when I walk that way.

Your original sentence is already naturally understood as “I take/wear them because it’s cold”, but these versions remove any ambiguity.

Why is vintern in the definite form, but mössa and halsduk are not?

Different reasons:

vintern
With på + season, Swedish almost always uses the definite form:
- på vintern, på sommaren, på våren, på hösten It means “in (the) winter / in winter” in general.
mössa, halsduk
When talking about clothes you put on yourself (your outfit at that moment), Swedish often uses them without article:
- Jag har mössa och halsduk. – I’m wearing a hat and scarf.
- Jag tar mössa och halsduk. – I take a hat and scarf (for myself).

So the forms are different because vintern follows the fixed seasonal expression, while mössa / halsduk follow the clothing-outfit pattern where the article is dropped.