Jag kan inte hälsa på henne i dag, för jag är fortfarande lite yr och hostar.

Because hälsa på is a fixed expression meaning to visit someone.

• hälsa on its own usually means to greet or to say hello
• hälsa på någon means to go and see/visit someone

So:

• Jag kan inte hälsa på henne = I can’t visit her
• Jag kan inte hälsa henne would sound wrong for this meaning

This is one of those verb + particle combinations you need to learn as a unit.

In this sentence, på is part of the expression hälsa på. It is not really functioning like the English preposition on.

So you should understand:

• hälsa på = visit
• hälsa på henne = visit her

A learner might try to translate word by word, but that does not work well here. Think of hälsa på as one vocabulary item.

Because henne is the object form of she/her.

• hon = she (subject form)
• henne = her (object form)

Here, she is the person being visited, so Swedish uses the object form:

• Jag kan inte hälsa på henne = I can’t visit her

Compare:

• Hon är sjuk = She is sick
• Jag ser henne = I see her

Because kan is the finite verb, and in a normal main clause Swedish usually puts inte after the finite verb.

Pattern:

• subject + finite verb + inte
  • rest

So:

• Jag kan inte hälsa på henne
• literally: I can not visit her

This is very common with modal verbs:

• Jag vill inte gå = I don’t want to go
• Hon måste inte stanna = She doesn’t have to stay / depending on context

kan is the present tense of kunna, which often means can / be able to.

So:

• Jag kan inte hälsa på henne = I can’t visit her

In Swedish, modal verbs like kan, vill, måste, and ska are followed by the infinitive form of the next verb:

• kan hälsa
• vill gå
• måste arbeta

That is why you get kan ... hälsa.

Yes. Both i dag and idag are used.

• i dag is the traditional two-word spelling
• idag is also very common in modern Swedish

They mean the same thing: today.

You may also see:

• i morgon / imorgon = tomorrow

So this is mostly a spelling/style issue, not a grammar difference.

Here för means because.

So:

• ..., för jag är fortfarande lite yr och hostar
• ..., because I am still a little dizzy and coughing

This is not the English preposition for. Swedish för can have several meanings depending on context, but in this sentence it is a conjunction introducing a reason.

A useful point: after för in this meaning, Swedish keeps normal main-clause word order:

• för jag är ...
• not something like a special subordinate-clause order

fortfarande means still.

In this sentence:

• jag är fortfarande lite yr
• I am still a little dizzy

With är, it is very natural to place fortfarande after the verb:

• subject + är
  • fortfarande
    • complement

Examples:

• Hon är fortfarande trött = She is still tired
• Jag är fortfarande sjuk = I’m still sick

So the placement here is completely normal.

lite means a little and yr means dizzy, light-headed, or sometimes giddy depending on context.

So:

• lite yr = a little dizzy

In this sentence, yr is an adjective describing how the speaker feels:

• jag är lite yr = I’m a little dizzy

Because Swedish can leave it out when the same subject continues for two coordinated verbs.

So:

• jag är fortfarande lite yr och hostar
• literally: I am still a little dizzy and cough

The subject jag applies to both är and hostar.

You could say:

• ..., för jag är fortfarande lite yr och jag hostar

but that sounds more repetitive. The version without the second jag is more natural here.

Because Swedish often uses the present tense for something happening now or currently true.

• hostar = am coughing / cough

In English, we often prefer I’m coughing in this kind of context, but Swedish simply uses the present tense:

• Jag hostar = I cough or I’m coughing, depending on context

So hostar works perfectly here to describe a current symptom.

No. yr is still an adjective.

After verbs like är, Swedish often uses an adjective to describe the subject. This is called a predicative adjective.

So in:

• jag är lite yr

yr describes jag, so it is an adjective, not an adverb.

Compare:

• Hon är glad = She is happy
• Jag är trött = I am tired
• Jag är yr = I am dizzy

Yes, you could. For example:

• Jag kan inte hälsa på henne i dag, eftersom jag fortfarande är lite yr och hostar.

That also means I can’t visit her today, because I’m still a little dizzy and coughing.

But there is a grammar difference:

• after för meaning because, Swedish keeps normal main-clause word order:
  för jag är fortfarande ...
• after eftersom, Swedish uses subordinate-clause word order:
  eftersom jag fortfarande är ...

So both are possible, but the word order changes.