Measurement of current and measurement of momentum of the electrons

Current is measure of average momentum(or velocity $v_d$) gained by the electron due to applied Electric field in a very short period of time($tau$) between the collision with other electrons or ions in the lattice.

Since at room temperature the electrons are very frequently undergoing collisions, the electrons can described only as statistical ensemble due to decoherence.

Yes the electrons occupy momentum eigen states with some probabilities which are given by the distribution function which depends on temperature.