On the basis of microscopic approach we derive the Eilenberger-type equations of superconductivity for metals with exchange-split conduction band. The equations are valid for arbitrary band splitting and arbitrary spin-dependent electron mean free paths within the quasiclassic approximation. Next, we deduce general boundary conditions for the above equations. These boundary conditions take into account explicitly spin-dependence of F/S interface transparency. We apply our theory for the Andreev reflection at F/S interface and derive an expression for the Andreev conductance at zero bias. Based on experimental data and our calculations we give estimations of the conduction band spin polarization for series of ferromagnets in contact with superconductors. Next, we consider the superconducting proximity effect for a contact of a strong and clean enough ferromagnet with a dirty superconductor. Our calculations show that superconducting Tc of an F/S bilayer oscillates as a function of the F-layer thickness. At small enough superconducting layer thickness the re-entrant behavior of superconductivity is predicted. The theory gives also nonmonotonic dependence of the superconducting layer critical thickness on the spin-polarization of the ferromagnetic layer. These unconventional and distinctive features of the F/S proximity effect fit well experimental observations.