biga ogi ttaemune usaneul gajyeowasseoyo.

Korean uses the nominalizer -기 to turn verbs into noun clauses so you can attach particles like 때문에.
• 오다 (to come) → 오기 (the coming/raining)
• 오기 + 때문에 = “because it is raining”

Yes. All three express reason, but with subtle differences:
• -아서/어서 (e.g., 비가 와서) is very common in spoken Korean and casually links clauses.
• -니까 (e.g., 비가 오니까) also gives a reason but can sound more direct or explanatory.
• -기 때문에 is slightly more formal or emphatic about cause and effect.

• 가져오다 = “to bring (to here)”
• 가져가다 = “to take (to there)”
Since the speaker brought an umbrella to the current location, 가져왔어요 is the correct choice.

Yes. 가지고 오다 and 가져오다 both mean “to bring.”
• 가지고 오다 is a bit more formal/literal (“to come holding/with something”), while
• 가져오다 is slightly more compressed and common in everyday speech.

Yes, colloquial speech often drops the verb:
• 비 때문에 우산을 가져왔어요 (“Because of rain, I brought an umbrella.”)
This works fine, though it relies on 비 as a noun rather than describing the action “raining.”