Learn Farsi: در اثر آن جیغ ترسناک، سگ نگهبان پارس کرد. - As a result of that scary scream, the guard dog barked.

dar asar-e means "as a result of" or "due to the effect of." While be khâter-e is a general "because of" or "for the sake of," dar asar-e specifically highlights a direct physical or situational cause-and-effect relationship. The scream was the direct catalyst for the bark.

Why does ân (that) come before the noun while tarsnâk (scary) comes after?

Demonstrative words like in (this) and ân (that) always precede the noun they modify and do not take the ezâfe linking vowel. Descriptive adjectives like tarsnâk, however, always follow the noun and must be connected to it with the ezâfe, which gives us the word order ân jigh-e tarsnâk.

Is negahbân in sag-e negahbân functioning as an adjective?

negahbân is actually a noun meaning "guard" or "watchman." Linking two nouns with the ezâfe (sag-e negahbân, literally "dog of guard") is the standard Farsi way to create compound concepts, working exactly like English noun combinations (guard dog, water bottle, bus stop).

How could we use this sentence's action with the interrupting ke structure taught in this lesson?

Because this sentence describes a sudden, completed event in the Simple Past, it functions perfectly as the "interruption" half of the structure. You would just need to add a background action in the Past Continuous before it. For example: dâshtim rad mishodim ke dar asar-e ân jigh, sag pârs kard (We were passing by when, due to that scream, the dog barked).