keompyuteo hwamyeoni jakgi ttaemune nuni apayo.

-기 is a nominalizer that turns the adjective 작다 into a noun-like element, allowing you to attach 때문에, which means because or due to.
Structure:
• 컴퓨터 화면이 작기 (the computer screen’s being small)
• 때문에 (because)
Together they form a reason clause: “Because the computer screen is small…”

You can’t directly attach 때문에 to the dictionary form 작다. You need to turn it into a noun phrase with -기.
• 작다 = “to be small” (adjective)
• 작기 = nominalized “smallness” or “being small”

Yes. All three express cause, but differ in formality and nuance:
• 작아서 (stem + 아/어서) = casual spoken style, simple “because…”
• 작으니까 (stem + 으니까) = casual, can imply a bit stronger or subjective reason
• 작기 때문에 = neutral to formal, more explicit “due to…”

-이/가 marks the grammatical subject or experiencer of a descriptive verb. Here, 화면이 is the subject of 작다, and 눈이 is the subject of 아프다.
If you use 은/는, you turn them into topics, shifting the focus:
• 화면은 작기 때문에… (As for the screen, because it’s small…)
• 눈은 아파요. (As for the eyes, they hurt.)

아프다 is a descriptive verb meaning “to hurt.” You say 눈이 아파요 (“the eyes hurt”).
아픈 눈이에요 would be an adjective + noun construction meaning “(it’s) a hurting eye,” which isn’t how Koreans describe pain.

Yes.
• 컴퓨터 모니터 = the physical monitor device
• 화면 = the screen/display itself
Both convey the idea; choose based on whether you mean hardware (모니터) or display area (화면).