If starting speed is faster than terminal velocity then what? If an object is say thrown down (vertically) at an initial speed that is faster than its terminal velocity, what would happen to that objects speed? Would it slow down?
 A: Terminal velocity results from a force balance between the falling body and the drag force acting upon it to slow it down. In other words, the terminal velocity is the speed at which the gravitational force is equal to the drag force (assuming the body is unpowered). 
So, if the initial velocity is larger than the terminal velocity, the drag force will be larger than the gravitational force. This results in a net force upwards, which results in an acceleration due to Newton's second law, and the body will slow down. Once it reaches terminal velocity, the forces are in balance and there is no more acceleration.
A: Yes.  The object will slow down to its terminal velocity if its speed starts higher than its terminal speed.  
The net force on a falling object of mass $m$ near the surface of the earth is
\begin{align}
  F = F_\mathrm{drag} - mg
\end{align}
where $F_\mathrm{drag}$ is the force due to air resistance, I have assigned "up" to be the positive direction, and $g$ is the magnitude of the acceleration due to gravity near the Earth's surface.  When the drag force is greater than $mg$, the net force will be positive, and by Newton's Second Law, the object's acceleration will point upward and will slow the object down.  This will reduce the drag force until the drag force and weight are equal, the object's acceleration is zero, and the object will have constant speed -- it's terminal speed.
