First, is there a way to determine the stiffness of the conical nozzle or input the conical portion of the tank through CAESAR II?
CAESAR II will only determine nozzle stiffnesses for WRC297, API650, and PD5500 nozzles. For other types of nozzles, you'll need to obtain the stiffness values from some other source (the vendor, or FEA).
You can't model a conical section in
CAESAR II. Everything is a "3D Beam", of constant cross section.
Second, if CAESAR II is not able to model this conical tank, is there a accurate means to calculate this by hand?
Not that I'm aware of. I would suggest the FEA route.