If I use {Jace, Architect of Thought}s last ability and leave the selected cards in exile, can I cast them if he is destroyed?
By the time Jace is destroyed, it would likely be too late anyway. You cast them as the ability resolves. No time frame is given which means that you are being told to cast them immediately. Leaving them in exile after the ability resolves means it's too late, whether or not Jace is destroyed.