Heavily electron-doped surfaces of Bi$_2$Se$_3$ have been studied by spin and angle resolved photoemission spectroscopy. Upon doping, electrons occupy a series of {\bf k}-split pairs of states above the topological surface state. The {\bf k}-splitting originates from the large spin-orbit coupling and results in a Rashba-type behavior, unequivocally demonstrated here via the spin analysis. The spin helicities of the lowest laying Rashba doublet and the adjacent topological surface state alternate in a left-right-left sequence. This spin configuration sets constraints to inter-band scattering channels opened by electron doping. A detailed analysis of the scattering rates suggests that intra-band scattering dominates with the largest effect coming from warping of the Fermi surface.