Learn Hardcore Norwegian: Katten sitter innenfor døren og ser ut. - The cat sits inside the door and looks out.

sitter (is sitting), står (is standing), ligger (is lying). They double as continuous forms, so Katten sitter … naturally means “The cat is sitting …”. You could say Katten er innenfor døren, but it’s less specific and less idiomatic when you can indicate posture. Using sitter also pairs well with another simultaneous action: … og ser ut.

What exactly does ser ut mean here, and how is it different from ser, ser på, and the “looks like” meaning?

Here ser ut = “looks out(ward).” It’s a particle verb (se + ut) expressing direction of gaze.
se (på) = “look at, watch.” Example: Katten sitter og ser på fuglene.
se etter = “look for.” Example: Jeg ser etter nøklene.
se ut som / se ut til (å) = “look like / appear to.” Example: Det ser ut som regn. / Det ser ut til å regne. In your sentence, without any complement, ser ut is understood as “is looking out (through the door/window to the outside).”

Why ut and not ute?

Norwegian distinguishes direction vs location:

ut = out(ward), directional.
ute = outside, location/state. With a verb of directed perception/action, you use ut: ser ut. By contrast: Katten er ute = “The cat is outside.”

What nuance does innenfor add compared to inne, inne i, or just i?

innenfor highlights being within a boundary, often right inside a threshold/barrier (door, gate, fence). Innenfor døren ≈ “just inside the door.”
inne = “inside/indoors” in general. Katten sitter inne is vaguer.
inne i emphasizes inside of something as a container/space: inne i huset (“inside the house”).
i = neutral “in.” So innenfor døren gives a near-threshold feel: the cat is just inside, at the doorway, not deep inside the room.

Could I use foran døren, ved døren, or bak døren instead? What would they mean?

foran døren = “in front of the door” (on the side the door faces; could be inside or outside depending on context).
ved døren = “by/next to the door” (close to it, side position).
bak døren = “behind the door” (hidden by the door leaf or on the opposite side). Your original innenfor døren specifically places the cat on the inside side of the doorway.

Where does ikke go if I want to negate, and how do I avoid the idiom ser ikke ut (“looks awful”)?

Standard placement: after the finite verb: ser ikke ut.
But beware: ser ikke ut is a common idiom meaning “looks terrible/awful.” To say “isn’t looking out,” add a complement so the meaning is unambiguously directional:
Katten sitter innenfor døren og ser ikke ut av vinduet/døra.
Or rephrase: Katten sitter innenfor døren uten å se ut. Don’t say ser ut ikke (wrong order).

Why do we need og between the two verbs? Can I say sitter ser ut?

You need og to coordinate two finite verbs with the same subject when describing simultaneous actions: sitter og ser. This construction is very common for ongoing activities (e.g., står og snakker, ligger og leser). Sitter ser ut is ungrammatical. Norwegian does not use an English-like progressive (“is sitting and looking”) or a chained bare-verb construction here.

Why are katten and døren in the definite form, and what about døra?

Norwegian marks definiteness with a suffix:

en katt → katten (the cat).
en/ei dør → døren/døra (both are standard Bokmål; døra is more colloquial; Nynorsk prefers døra). You use the definite because you’re referring to a specific, known cat and door. With adjectives, you’d add a separate article: den svarte katten, den gamle døren/døra.

Can I front the place phrase: Innenfor døren sitter katten og ser ut?

Yes. This is good Norwegian and slightly emphasizes location. It also illustrates the V2 rule: in main clauses, the finite verb (sitter) must be in second position, so after fronting Innenfor døren, the verb comes before the subject: Innenfor døren sitter katten …

Pronunciation tips for the sentence?

katten: [ˈkɑtːən] (double t = long/consonant geminate).
sitter: [ˈsɪtːər].
innenfor: [ˈɪnːənfɔr] (double n).
døren: [ˈdøːrən] (ø like French eu in “peur”; r is tapped).
ser: [seːr] (long e).
ut: [ʉːt] (u is a fronted vowel, like German ü). Stress each content word on the first syllable: KAT-ten SIT-ter IN-nen-for DØ-ren SER UT.

How can I make the “out where?” more explicit, e.g., window vs door vs view?

Add a complement after ut:

Through/from: ser ut av vinduet / ut døra / ut gjennom vinduet (all common; ut av is most idiomatic for “out of”).
Toward/on(to) a view: ser ut mot havet, ser ut på gata. These clarify direction and also prevent the idiom ser ikke ut when negated.

Can I swap sitter for other posture verbs?

Yes, to reflect posture:

Katten ligger innenfor døren og ser ut. (is lying)
Katten står innenfor døren og ser ut. (is standing; less typical for a cat, but grammatical) Norwegian uses these posture verbs frequently to paint a more precise picture than plain er.