You are doing well here!
In the first case (which I also have since my grandfather married twice only my mother is from the second marriage) he is your uncle (or half-uncle if you want to be technically accurate). He isn't your cousin but any children he has are or will be your cousins.
In the second situation you are correct...second cousins once removed. Removed comes into play when the number of generations between the common ancestor and you and the cousin are not the same. Each generation added to one side (but not the other) is another generation removed for the cousin relationship.