Hello, Ms.Tara! Actually, "失くす" and "無くす" are almost same meaning, "lose". For example, 「私は鍵を無くした（失くした）」 means " I lost my key." On the other hand, "亡くす" is special word related with death. For example, 「私は父を亡くした」 means " I lost my father." In that meaning, we can't say 「私は親を無くした（失くした）」.

"壊す" means "break". For example, 「私は傘を壊した」 means " I broke an umbrella." On the other hand, "壊れる" means " broken". For example, 「私の傘は壊れた」 means " My umbrella was broken." "壊す" is used in positive voice and "壊れる" in passive voice.