Learn Hardcore Norwegian: Fakturaen er allerede betalt, så jeg kan slappe av i kveld. - The invoice is already paid, so I can relax this evening.

Fakturaen er betalt = The invoice is paid (focus on the state/result; who paid is not important or not mentioned).
Jeg har betalt fakturaen = I have paid the invoice (active; you explicitly say who paid).

So er betalt functions like a result/state description (often similar to a passive/resultative meaning in English).

What exactly is betalt, and why is it not betalte?

Betalt is the past participle of å betale (to pay). In er betalt, it’s used like an adjective describing the invoice’s status.

You don’t use betalte here because betalte is typically the plural/definite adjective form:

en betalt faktura / fakturaen er betalt (singular)
betalte fakturaer / fakturaene er betalte (plural)

So fakturaen er betalt matches singular.

Where can allerede go, and does its position matter?

Allerede (already) is flexible, but some positions are more natural than others.

Common/natural:

Fakturaen er allerede betalt.
Fakturaen er betalt allerede. (often a bit more emphatic, like already as in sooner than expected)

In general, putting allerede before the past participle (betalt) is very common.

Why is there a comma before så?

Because it’s connecting two independent clauses: 1) Fakturaen er allerede betalt 2) jeg kan slappe av i kveld

In Norwegian, you normally use a comma before coordinating conjunctions like så when they join two full clauses (similar to English in many cases).

Why is the word order så jeg kan ... and not så kan jeg ...?

Here så means so (a conjunction), and after conjunctions you use normal main-clause word order: subject before the verb:

..., så jeg kan slappe av i kveld.

If så means then (an adverb) and comes first in a clause, it triggers inversion (verb before subject):

Så kan jeg slappe av i kveld. = Then I can relax tonight.

So the meaning/role of så affects word order.

Could I replace så with something else?

Yes, depending on nuance:

..., så ... = so (neutral, common)
Derfor kan jeg slappe av i kveld. = Therefore I can relax tonight. (a bit more formal)
..., sånn at jeg kan slappe av i kveld. = so that I can relax tonight (more purpose-focused; slightly different structure)

Your original sentence is the most straightforward.

How does kan work here, and where would ikke go?

Kan is a modal verb (can/be able to). Modals are followed by an infinitive:

kan slappe av = can relax

If you negate, ikke usually comes after the modal:

..., så jeg kan ikke slappe av i kveld. = so I can’t relax tonight

Is slappe av one verb, and can it be separated?

Yes—å slappe av is a common verb + particle combination (relax). It often behaves like a separable phrasal verb:

Infinitive (stays together): å slappe av, kan slappe av
Finite verb in simple tenses (often splits): Jeg slapper av. / Jeg slapp av i går.
With an object in between (possible): Jeg slappet meg av is not standard; instead you’d rephrase. Normally slappe av doesn’t take a direct object like that.

In your sentence, it stays together because it’s an infinitive after kan.

Why is it i kveld and not i kvelden?

I kveld is the standard expression for tonight (like a fixed phrase).

I kvelden is generally not used to mean tonight in normal modern Norwegian. If you say kvelden (the evening), it usually sounds like you’re referring to a specific evening as a known time period, and you’d typically phrase it differently (e.g., den kvelden, på kvelden, etc.).