Ég tek af mér úlpuna áður en ég fer inn í húsið.

Because tek is the 1st person singular present-tense form of taka (to take).

Present tense of taka:

• ég tek = I take
• þú tekur = you take
• hann/hún/það tekur = he/she/it takes

So with ég, you must say tek.

Literally, it is something like take off myself or take from myself.

This is a very common Icelandic way to say take off an item of clothing:

• taka af sér = to take off, remove from oneself

So in this sentence:

• Ég tek af mér úlpuna = I take off my coat/jacket

English uses a single phrasal verb, take off. Icelandic often uses taka af plus a pronoun showing who the item is being removed from.

Because the expression taka af takes the pronoun in the dative case, not the accusative.

So you get:

• af mér = off me / from me
• af þér = off you / from you
• af sér = off himself/herself/themselves

That is why mig is not used here. Mig is accusative, but this construction requires dative.

In meaning, yes: it refers back to the subject ég.

But grammatically, Icelandic only has a special reflexive pronoun in the 3rd person: sig / sér / sín.
For 1st and 2nd person, ordinary pronouns do the same job:

• Ég tek af mér úlpuna = I take off my coat
• Þú tekur af þér úlpuna = You take off your coat
• Hann tekur af sér úlpuna = He takes off his coat

So mér is not a special reflexive form, but it functions reflexively in the sentence.

Because úlpuna is the accusative singular definite form of úlpa.

Here, úlpuna is the direct object of tek:

• Ég tek ... úlpuna = I take off the coat/jacket

Forms of úlpa:

• úlpa = a coat/jacket (nominative singular)
• úlpu = a coat/jacket (accusative singular indefinite)
• úlpuna = the coat/jacket (accusative singular definite)

So the sentence uses úlpuna because it means the coat/jacket and because it is the object of the verb.

Because af mér is closely tied to the verb phrase taka af sér.

Icelandic often keeps this kind of small pronoun phrase near the verb, especially in common expressions. So:

• Ég tek af mér úlpuna sounds very natural.

You may also hear or see:

• Ég tek úlpuna af mér

But taka af sér is such a common pattern that many learners find it easiest to understand it almost as one unit:

• tek af mér = take off myself

Áður en means before when it introduces a full clause.

So:

• áður en ég fer inn í húsið = before I go into the house

It is a conjunction made of two words:

• áður = earlier, before
• en = than / when used in certain conjunctions

Together, áður en introduces something that happens earlier than the action in the main clause.

Because Icelandic, like English, often uses the present tense in clauses like this for something habitual, general, or near-future in meaning.

So:

• áður en ég fer inn í húsið literally uses present tense
• natural English is also before I go into the house

This sentence can describe a regular action:

• I take off my coat before I go into the house

So the present tense here is completely normal.

Because inn adds the idea of inward movement, specifically into the inside of something.

• í húsið = into the house / to the house, depending on context
• inn í húsið = into the house, clearly emphasizing entry inside

So inn makes the meaning more explicit:

• fer inn í húsið = go into the house

In everyday Icelandic, this is a very common way to say that someone is going inside.

Because the preposition í takes:

• accusative when there is movement toward/into something
• dative when there is location in something

Here there is movement:

• ég fer inn í húsið = I go into the house

So the accusative is used:

• húsið

Compare:

• Ég fer inn í húsið. = I go into the house.
• Ég er inni í húsinu. = I am inside the house.

That contrast is very important in Icelandic.

They are related, but they are used differently:

• inn í = into, inward movement
• inni í = inside, location

Examples:

• Ég fer inn í húsið. = I go into the house.
• Ég er inni í húsinu. = I am inside the house.

So in your sentence, inn í is used because the speaker is moving into the house, not already located there.

Yes. Icelandic word order is flexible, especially for emphasis or style.

For example, you could also say:

• Áður en ég fer inn í húsið, tek ég af mér úlpuna.

That means the same thing, but it puts the before-clause first.

Notice that when the subordinate clause comes first, the verb comes before the subject in the main clause:

• tek ég not
• ég tek

That is because Icelandic follows the V2 pattern in main clauses: the finite verb usually comes in the second position.