Some questions about using h-parameters to model a BJT
But BJT is a voltage controlled current source which means Vbe must be the independent variable.



This is your bias (input voltage). Many users put emphasis on input current rather than voltage. Even so, current gain for many BJT is not constant over the whole universe of operating current. As for output, a current source is an obvious choice. Nothing stops you from characterizing BJT with H-parameters for common-base or common-collector.
H-parameters only apply to linear circuits. Can't use them when output is not proportional to input. For example, a sine wave going in should yield a sine wave coming out of the 2-port.