Hvort sem ég fer í stuttermabol eða regnfrakka, tek ég alltaf hettupeysu með mér í vinnuna.

It is a fixed pattern meaning whether ... or ....

So:

Hvort sem ég fer í stuttermabol eða regnfrakka ...
= Whether I wear/put on a T-shirt or a raincoat ...

It sets up two alternatives, and the main clause is true in both cases. In other words, no matter which of the two happens, the result is the same.

The word sem is just part of this expression here; you do not need to translate it separately word by word.

In Icelandic, að fara í is very commonly used with clothing and means to put on or, in many contexts like this one, to go wearing.

So:

• fara í stuttermabol = put on / wear a T-shirt
• fara í regnfrakka = put on / wear a raincoat

In this sentence, the most natural understanding is something like whether I go in a T-shirt or a raincoat or simply whether I wear a T-shirt or a raincoat.

English usually uses wear, but Icelandic often uses fara í in this kind of context.

It can suggest either, depending on context.

Literally, að fara í föt often means to put clothes on. But in a sentence like this, where the point is what kind of clothing the speaker ends up in, English usually translates it more naturally as wear:

• Hvort sem ég fer í stuttermabol eða regnfrakka ...
• Whether I wear a T-shirt or a raincoat ...

So the Icelandic expression is a little broader than a single English verb.

Because they are in the accusative case.

After fara í when it means put on / go into, Icelandic uses the accusative. So the nouns change form:

• stuttermabolur → stuttermabol
• regnfrakki → regnfrakka

These are the accusative singular forms.

So this is not random spelling change; it is normal Icelandic case inflection.

This is because of Icelandic verb-second word order in main clauses.

The first part of the sentence, Hvort sem ég fer í stuttermabol eða regnfrakka, comes first. After that, the main clause begins, and the finite verb must come early:

• tek ég alltaf ...

If you wrote the main clause by itself, it would normally be:

• Ég tek alltaf hettupeysu með mér í vinnuna.

But when something else is placed first, Icelandic usually moves the verb before the subject in the main clause.

This is very normal and very important in Icelandic syntax.

Með mér means with me.

The phrase að taka eitthvað með sér means to take something with you / to bring something along.

In this sentence:

• tek ... með mér
• I take ... with me

Because the subject is ég, the form is mér.

Compare:

• ég tek með mér = I take with me
• þú tekur með þér = you take with you
• hann tekur með sér = he takes with him
• við tökum með okkur = we take with us

So með mér is not extra or optional fluff; it is a very common Icelandic way to express bring along / take with you.

Because með takes the dative case.

The pronoun ég has these forms:

• nominative: ég
• accusative: mig
• dative: mér
• genitive: mín

Since með requires dative, Icelandic uses mér:

• með mér = with me

So með mig would be wrong here.

Because hettupeysa is the direct object of tek, so it is in the accusative singular.

The noun is:

• nominative: hettupeysa
• accusative: hettupeysu

So:

• Ég tek hettupeysu ...
• I take a hoodie ...

This is a very common pattern with feminine nouns ending in -a: they often become -u in the accusative singular.

Here í vinnuna means to work or more literally to the workplace.

The important grammar point is:

• í + accusative often shows motion toward a place
• í + dative often shows location in a place

So:

• í vinnuna = to work, into the workplace
• í vinnunni = at work, in the workplace

Because the speaker is taking the hoodie along to work, the sentence uses í vinnuna.

Because the clothes are being mentioned indefinitely, while vinnuna refers to a specific, familiar destination.

So:

• stuttermabol = a T-shirt
• regnfrakka = a raincoat

But:

• vinnuna = the work / the workplace

In Icelandic, routine destinations like home, school, or work are often expressed in ways that do not match English word-for-word. So í vinnuna is very natural even though English usually just says to work without the.

Yes. That is often the most natural English translation.

Literally, Icelandic says:

• I take a hoodie with me to work

But idiomatic English may say:

• I bring a hoodie to work

So if you were wondering why Icelandic uses take where English might prefer bring, that is normal. The Icelandic wording is perfectly natural.

Alltaf means always, and its position here is very normal.

In a simple main clause, you might say:

• Ég tek alltaf hettupeysu með mér í vinnuna.

After the fronted opening clause, Icelandic keeps the verb-second structure:

• Hvort sem ... , tek ég alltaf hettupeysu ...

So the order becomes:

1. fronted clause
2. finite verb: tek
3. subject: ég
4. adverb: alltaf
5. object and the rest

That placement of alltaf is standard and natural.