I'm afraid you are not going to like my answer.
I'm a CGAL developer, actually one of the developers of the Reg. Boolean Operation and the Arrangement packages.
First, the operations you are asking for are not supported.
Regarding the area computation, your approach seems feasible. It will be, however, a great effort for us to require such an operation in the concept, because then we would need to implement the operation for all the traits classes we support. I guess, it is always possible to start with one and then adding them one by one. I will throw it into our todo list, but I wouldn't put my bets on a quick delivery...
Regarding the transformation, the answer is more involved. As you have already noticed, applying a non-exact transform to the (exact) geometric shape (e.g., arrangement, general polygon set, or even a small linear simple convex polygon) can be harmful. You must come up with an exact transform, for example, a transformation matrix that comprises numbers of an exact type---the same (or at least convertible to one another) type used to represent the (exact) coordinates of the geometric elements. The problem is, naturally, rotation, because you typically start with an angle and use trigonometric functions (e.g., sin() and cos()) to compute the rotation matrix. Let's say you want to rotate by a given angle, say alpha. You need to compute an approximation of alpha, such that sin(alpha) and cos(alpha) are rational numbers, and thus can be represented by numbers of the aforementioned exact type. The free function CGAL::rational_rotation_approximation() can help. As mentioned in the manual entry of this function, the approximation is based on Farey sequences as described in the rational rotation method presented by Canny and Ressler at the 8th SoCG 1992.
Good luck!