Angular and linear velocity of a circle 
Let's assume I have a circle of radius $r$ and I apply a constant force at a point on the circumference of the circle that is perpendicular to the distance vector which goes from the centre of the circle to the point. If I wanted to find how fast it is rotating (angular velocity) at a specific time, I can integrate the acceleration vector ($F/m$) to get the velocity vector and then do the cross product between the distance vector from the centre to the point and the velocity vector (and divide by $r^2$) to find the angular velocity. Now if the circle is tied down (or fixed) at it's centre, it will simply rotate in place. If it's not fixed, would it be correct to assume that on top of spinning around it's centre of mass, the centre of mass will also be moving? If so, how can I find the velocity at which the centre of the circle is moving (not rotating). We're not talking about a circle that is rolling like wheels, we're talking about a circle that is flat with the ground and we can assume that the circle has no friction.
 A: The center of mass moves with an acceleration given by $\vec{F} = m \vec{a}$, just as if it were a point mass.  You can then integrate $\vec{a}$ to find the position of the center as a function of time.
I should also note that the method you describe will only work for a hoop where all the mass is concentrated at a fixed radius $r$.  It will not work for a solid disk, for example.  The better method is to find the torque exerted by the force about the center of mass $\tau$, use Newton's Third Law for rotations ($\tau = I \alpha$) to find the angular acceleration, and then integrate that to find the angular velocity as a function of time.
A: This is a problem of rigid body mechanics. The movement can be separated into trasnlational motion and rotational motion.
Translation of the center of mass.
The center of mass moves LIKE if all forces were acting on it. No matter where the forces are, the center of mass moves like if all forces were acting on the center of mass.
Then, in addition, you can have motion around the center of mass. In fact, you can describe rotation around any point, but it is usual to refer rotation to the center of mass, and your particular geometry is quite suitable for it.
To know how it rotates around the center of mass, you need to calculate the torque of the force w.r.t. the center of mass. Then, use the formula $\tau=I\cdot \alpha$, where $I$ is the moment of inertia of the disk, $\alpha$ is the angular acceleration, and $\tau$ is the torque.
