Я забыл шарф дома, поэтому мне холодно.

Забыл is past tense of the perfective verb забыть and refers to a single completed event: you ended up not having the scarf because you forgot/left it behind.
Забывал (imperfective, забывать) would usually mean something like used to forget (repeatedly) or would focus on the process/background rather than the completed result.

In this structure, it means you left it at home / forgot to take it: забыть + object + location → забыть шарф дома = to forget/leave the scarf at home.
If you want “forgot about the scarf,” Russian typically uses забыть о + prepositional: Я забыл о шарфе.

Because шарф is a masculine noun and here it’s the direct object of забыл, so it’s in the accusative case. For many masculine inanimate nouns, the accusative equals the nominative: шарф → шарф.

дома is an adverb meaning at home (a fixed common form). It’s the most natural choice for “at home” in everyday speech.
в доме literally means in the house and sounds more specific/physical (inside the building), not “at home” as a general situation.

поэтому means therefore / so / that’s why and introduces the result.
When it connects two independent parts of a sentence (two clauses), Russian normally uses a comma:
Я забыл шарф дома, поэтому мне холодно. = I left my scarf at home, so I’m cold.

Russian often expresses physical states with an impersonal construction:
мне холодно = to me (it is) cold → I’m cold.
я холодный usually describes a characteristic or tactile quality: I am a cold person / my body feels cold to the touch (or metaphorically “emotionally cold”).

In мне холодно, the person experiencing the state is marked with the dative: кому? → мне.
This is standard with many “state” words: мне жарко (I’m hot), мне скучно (I’m bored), мне плохо (I feel bad).

Yes, Russian word order is flexible, but it changes emphasis. For example:

• Я забыл шарф дома, поэтому мне холодно. (neutral)
• Дома я забыл шарф, поэтому мне холодно. (emphasis on at home)
• Мне холодно, потому что я забыл шарф дома. (starts with the result)

They point in opposite directions logically:

• поэтому introduces the result: I forgot my scarf; therefore I’m cold.
• потому что introduces the cause: I’m cold because I forgot my scarf.
  Both are correct; you just choose which clause you want to present first.

Yes, past tense agrees with the speaker’s gender:

• male speaker: Я забыл шарф дома...
• female speaker: Я забыла шарф дома...
  Everything else stays the same.

A practical guide to stress (stressed syllable in CAPS):

• Я забы́л шарф до́ма, поэ́тому мне хо́лодно.
  Notes: забы́л has stress on the second syllable; поэ́тому has stress on э́.