Learn Hardcore Norwegian: Du kan ringe når som helst hvis du blir syk. - You can call at any time if you get sick.

In Norwegian, modal verbs (like kan, må, vil, skal) are followed by the bare infinitive without the infinitive marker å and without the present-tense ending -r. So you say kan ringe, not kan ringer and not kan å ringe.

Does kan here mean ability or permission?

Both are possible, depending on context. In this sentence it most naturally gives permission: “you may/are allowed to call.” If you want to sound especially welcoming, you can say:

Du kan gjerne ringe ...
Du kan godt ringe ...
Bare ring ...

Do I need til after ringe? What’s the difference between ringe noen and ringe til noen?

ringe noen = call someone (very common and natural)
ringe til noen = also correct; slightly more formal/regional feel in some ears
Don’t confuse with ringe på = ring the doorbell (e.g., ringe på døra)

You can also use ringe without an object when the recipient is understood: Du kan ringe når som helst.

Where can I put når som helst? Is its position fixed?

It’s most natural at the end of the main clause: Du kan ringe når som helst ...
Other options:

Fronted for emphasis (V2 rule applies): Når som helst kan du ringe hvis du blir syk.
Midfield (acceptable but a bit marked): Du kan når som helst ringe hvis du blir syk.

Avoid splitting it awkwardly from what it modifies.

Should there be a comma before hvis?

No comma is used before a following subordinate clause here: Du kan ringe når som helst hvis du blir syk.
If the hvis-clause comes first, add a comma after it:

Hvis du blir syk, kan du ringe når som helst.

Why is it hvis du blir and not hvis blir du?

In subordinate clauses introduced by hvis, Norwegian uses normal subject–verb order (no inversion): hvis du blir.
Inversion (verb before subject) happens in main clauses after fronted elements due to the V2 rule, not inside subordinate clauses.

Could I use når instead of hvis here?

Not if you mean “if.”

hvis = if (a condition that may or may not happen)
når = when (used for time, and often implies it will happen)

Når du blir syk would suggest it’s expected to happen at some point, which is usually not the intended meaning.

What’s the difference between hvis and om?

Both can mean “if” in conditionals, and many speakers use them interchangeably there:

Du kan ringe ... hvis/om du blir syk.

However, om is also used for “whether” in indirect questions:

Jeg lurer på om du blir syk. (I wonder whether you’ll get sick.)
You can’t use hvis for “whether.” In careful writing, prefer hvis for conditions.

Why blir syk and not er syk?

blir syk = “get/become sick” (onset of illness; future/possible event)
er syk = “are sick” (current state)

So for a conditional about a future possibility, hvis du blir syk is natural. A more tentative variant is hvis du skulle bli syk (“if you should happen to get sick”).

How does the adjective syk work with different subjects?

Predicate adjectives agree in number and neuter:

Singular common gender: Han/Hun/Du er syk.
Neuter: Det er sykt.
Plural: De er syke.

Here, du takes the base form syk.

What does når som helst literally do, and where else is som helst used?

som helst adds the sense “-ever/any.” With question words it yields:

når som helst = any time/whenever
hvor som helst = anywhere
hvem som helst = anyone
hva som helst = anything
hvilken som helst = any (which one)

A near-synonym for time is til enhver tid (at any time).

How do I pronounce tricky parts like hvis, ringe, når, and syk?

Approximate guide (Eastern Norwegian):

hvis: the h is silent; sounds like “vis.”
ringe: “RIN-geh” (ng = [ŋ]).
når: “nohr” (long o).
syk: “syuk” with a long front rounded vowel [y] (like French u).

Don’t pronounce h in hv- words in standard Bokmål speech.

Can I drop the subject and say Kan ringe ...?

No. Norwegian is not a subject-dropping language in statements. You need du: Du kan ringe ...
If you want a command, use the imperative: Ring når som helst hvis du blir syk.

Where does the negation ikke go if I want to say “You can’t call ...”?

Place ikke after the finite verb kan and before the infinitive:

Du kan ikke ringe når som helst hvis du blir syk.

Are there more formal or nuanced variants of the sentence?

Yes:

More tentative/polite: Skulle du bli syk, kan du ringe når som helst.
Fronted time for emphasis: Når som helst kan du ringe hvis du blir syk.
Warmer invitation: Du kan gjerne ringe når som helst hvis du blir syk.