Wrong. Everything has some level of stiffness/flexibility. CF rods are very stiff, stiffer than wood so adding them increases the overall stiffness of the neck, but they are still flexible. Plus, their size affects their flexibility.
The complete neck will have some level of stiffness/flexibility. The string tension will serve to bow it forward to some degree, depending on how stiff it is; the truss rod is used to bow it back to a nearly equal degree, to obtain the commonly desired result of a very slight forward bow.
Again, a very stiff neck, whether it results from adding CF rods or making it out of purpleheart, is harder to bend. It is not dead stiff, though. Even a diamond neck would flex to some degree.
And CF rods are cemented in place. Most people use epoxy. As one can read in nearly every single build on this forum (some people don't use CF), or in every guitar building book.
