So let's say I have a {Silvergill Adept} on the board and throw a {Phantasmal Image}, copying it. Would I have to pay any additional casting costs since it enters as an Adept?
No. You are casting Phantasmal Image, and not the Merfolk. You don't need to pay additional costs. However, it enters the battlefield as the Merfolk thing, so you do draw a card. I cannot find an additional rule to cite this.