Is \$I_e = I_c\$ for phototransistors?
It must be; there is nowhere else for the charge to go.

Photons don't create electrons; they create electron-hole pairs. The electron is accelerated to the collector, the hole to the emitter (for NPN phototransistors; opposite for PNP ones).

In a photodiode, some of the electrons and holes end up on opposite sides of the barrier and can't easily recombine until (a) they build up a voltage of about 0.5V to attract them back across or (b) you provide an easier external current path. This is turning into quite a big industry...