why is the transverse modulus of carbon fibers lower than their longitudinal modulus?

Answer 1

The longitudinal modulus of a carbon fiber matrix is determined by loading the carbon fibers in a longitudinal direction, parallel to the longitudinal direction of the fibers, which is loading the carbon fibers across their strongest cross section, similar to pulling on a series of steel cables laid side by side.

When loading in a transverse direction what you are really measuring is the strength of the epoxy matrix that binds all of the carbon fibers together. You are pulling in a direction that lies at 90 degrees to the longitudinal direction.
This is equivalent to gluing a bunch of steel cables together in a bundle and then applying a load at 90 degrees to the longitudinal direction that tries to pull the cables apart by breaking the glue bond between the cables.
This transverse modulus is much bore brittle and is weaker than the longitudinal modulus.
To overcome some of this the carbon fibers are usually wound at an angle which then adds some of the fiber strength to the lateral direction.
The same applies to epoxy/glass fiber assemblies.