Learn Hardcore Norwegian: Vi tar av skoene i gangen og legger hanskene ved dørmatten. - We take off the shoes in the hallway and place the gloves by the doormat.

tar av literally means “take off.” It’s a separable verb made up of ta (to take) and the particle av (off). In the present tense, ta becomes tar, and you keep av right after it when the object is a noun:
Vi tar av skoene.
(We take off the shoes.)

Why isn’t there a reflexive pronoun seg as in tar av seg skoene?

You can include seg if you want to stress that the subject removes its own shoes:
Vi tar av seg skoene.
But when it’s clear whose shoes they are, Norwegians often drop seg and simply say tar av skoene.

Why are skoene and hanskene in the definite plural form, not just sko and hansker?

Norwegian marks definiteness by adding a suffix to the noun:
Indefinite plural sko → definite plural skoene
Indefinite plural hansker → definite plural hanskene
We use the definite form here because we’re talking about specific shoes and gloves (the ones you’re wearing).

Why don’t we say våre sko or våre hansker, since they belong to us?

If the possessor is obvious from context, Norwegian prefers to drop the possessive pronoun and just use the definite form of the noun. You could say våre sko (“our shoes”), but it’s redundant when it’s clear whose shoes they are.

What’s the difference between i gangen, på gangen, and ved dørmatten, and why are those prepositions used?

i gangen = “in the hallway,” inside that space.
på gangen can also mean “in the corridor,” but with a slight nuance of “on the hall surface.” Both are common, though i gangen is more typical for rooms/areas.
ved dørmatten = “by/next to the doormat.” If you said på dørmatten, it would mean “on top of the doormat.”

Can we move the location phrases around? For example, start with I gangen tar vi av skoene…?

Yes. Norwegian follows the V2 rule: one element comes first, then the verb, then the subject (if the first element wasn’t the subject). So you can say:
I gangen tar vi av skoene og legger hanskene ved dørmatten.
This puts emphasis on i gangen. Similarly, you could front ved dørmatten if you want to highlight where you leave the gloves.

How do you form the definite singular gangen and dørmatten, and what genders are these nouns?

Both gang (“hallway”) and dørmatte (“doormat”) are common-gender nouns (formerly called masculine or feminine).
en gang → gangen
en dørmatte → dørmatten
Common-gender nouns add -en for the definite singular.

Could we replace hanskene with a pronoun to avoid repetition, e.g. …og legger dem ved dørmatten?

Yes. You can say:
Vi tar av skoene i gangen og legger dem ved dørmatten.
Here dem refers to hanskene. Just be careful: if you use dem twice (e.g. tar dem av), it can get confusing which object you mean. Repeating the noun is often clearer.