Learn Italian: Avremo molti problemi per colpa di quello sbaglio. - We will have many problems because of that mistake.

Both mean "because of", but per colpa di specifically implies blame or a negative outcome, translating literally to "through the fault of". Since a mistake (sbaglio) is causing the problems, per colpa di is the most natural fit. A causa di is generally more neutral.

Could I just use perché instead of per colpa di to say "because of"?

No. Perché is a conjunction and must be followed by a subject and a conjugated verb (like "because he made a mistake"). When "because of" is followed directly by a noun phrase like quello sbaglio, you must use a prepositional phrase like per colpa di or a causa di.

Why is it molti problemi? Doesn't the singular word problema end in a feminine "a"?

Problema is actually a masculine noun, despite ending in -a. It is one of several Italian words of Greek origin (like il programma or il sistema) that are masculine. Therefore, its plural is i problemi, and the adjective must agree as the masculine plural molti.

Why does the sentence use quello sbaglio instead of quel sbaglio?