Why does this require a const_cast?

In short, to answer your first quesiton, yes, the const ness is essentially imbued to member variables (see here, in the structs example). The whole object is considered const. If you don't want this to be the case, you can mark one of your members as mutable.

Your second question is answered by the link above. Since the member variable is actually considered const, constcasting away its constness is undefined behavior, in the same way that this is:

const int i = 1;
const int& thisguy = i;
constcast(thisguy) = 2; // UB

I was under the impression that the implementer of the class could do whatever they wanted and that it was merely the case that const objects could only call const functions.

struct C
{
   int x;
   mutable int y;
   C()
   {
    x= 4; //this works even for const objects as we're inside ctor 
   }
   void changexy() const 
   {
    // x = 4; //this won't compile because x is not mutable 
    y = 4; //this will compile because y is marked mutable 
   }
};
const C c; //this uses the ctor which can change value of x as well as y
int main()
{
     c.changexy();
}