Doubts on Semiconductor Physics
(1) It is simply Kirchhoff's law $$I_E=I_B+I_C$$

(2) $I_C$ is (approximately) proportional to $I_B$ because both currents are determined by the base-emitter forward bias voltade $V_{BE}$ via the exponential proportionality factor (diode factor) $\exp (qV_{BE}/kT)$ $$I_C \propto I_B \propto \exp (qV_{BE}/kT)$$ The proportionality factor between collector and base current is called the dc current amplification factor. ($q$ is the electron charge, $k$ the Boltzmann constant, $T$ the absolute temperature)