I have been trying to figure out why does rotation about an arbitrary axis have degree of freedom 6. I know that it takes two points, each take 3 values to determine in a 3-D space. But what about the rotation angle? Wouldn't that make the degree of freedom increase by 1 to get 7?
It is only 3 degrees of freedom (when talking about a rotation around the origin).
An arbitrary axis can be specified by 1 3D vector however the length doesn't matter so it only takes 2 degrees of freedom. Then you add the rotation angle to it and you have 3 degrees of freedom.
If you mean a rotation around a arbitrary point other than the origin then that point takes up 3 additional degrees of freedom for a total of 6.