Learn Hardcore Norwegian: Når du er ferdig, ringer du meg.

Because the sentence starts with a subordinate clause (Når du er ferdig). In Norwegian main clauses, the verb normally comes in the second position (V2 rule).
So when the time clause comes first, the main clause still needs the verb in position 2, causing inversion:

Når du er ferdig, ringer du meg. (subordinate clause first → verb before subject)
Du ringer meg når du er ferdig. (main clause first → normal order)

What exactly is Når doing here? Could it be da instead?

Når introduces a time clause meaning when/whenever (often general or repeated situations, and also commonly used for future events).
Da is usually used for when (then) in a specific past situation:

General/future: Når du er ferdig, ringer du meg.
Specific past: Da du var ferdig, ringte du meg.

Why is the verb in the present tense (ringer) if the meaning is about the future?

Norwegian often uses the present tense to talk about future events, especially with a time expression like når:

Når du er ferdig, ringer du meg. = “When you’re done, you’ll call me.”

You can add a future marker, but it’s usually not necessary here.

Is er ferdig a tense (like present perfect), or what kind of construction is it?

It’s not a perfect tense. Er is the present tense of å være (to be), and ferdig is an adjective meaning finished/done/ready.
So du er ferdig literally means you are finished/done (i.e., you have completed what you’re doing).

Why is there a comma in Når du er ferdig, ringer du meg?

Norwegian normally uses a comma after an introductory subordinate clause:

Når du er ferdig, ringer du meg.

If the subordinate clause comes last, there is typically no comma:

Du ringer meg når du er ferdig.

Why does meg come after ringer? Can it go earlier?

In neutral word order, the object pronoun comes after the verb:

ringer du meg

You can move meg earlier for emphasis/contrast, but then the word order changes and it sounds marked:

Meg ringer du (når du er ferdig). = “Me, you call (when you’re done).”

Could I also say Når du blir ferdig instead of Når du er ferdig?

Yes, both can work, with a small nuance:

Når du er ferdig focuses on the state: when you are done.
Når du blir ferdig focuses a bit more on the transition/process: when you get finished.

In many everyday contexts they’re interchangeable.

Does this sound like a command? How would I make it more polite or more direct?

As written, it often sounds like an instruction in the form of a statement (common in Norwegian). Options:

More direct (imperative): Når du er ferdig, ring meg.
More polite: Når du er ferdig, kan du ringe meg?
Even softer: Kunne du ringt meg når du er ferdig?

Can I change the order and put the Når clause at the end?

Yes, and it becomes more like plain SVO word order:

Du ringer meg når du er ferdig.

Both are correct; putting the time clause first often highlights the timing/condition.

Any pronunciation pitfalls in Når, ferdig, or ringer?

Common ones:

Når: the å is like the vowel in British English thought (varies by dialect), and ng is a single sound like in sing.
ferdig: the g at the end is often soft or barely pronounced in many dialects (you may hear something like ferdi).
ringer: the -er ending is usually a reduced sound (not a clear English er), and ng again is like in sing.