The interplay between chain conformations and phase separation in binary symmetric polymer mixtures confined into thin films by "neutral" hard walls (i.e., walls that do not preferentially attract or repel one of the two components of the mixture) is studied by Monte Carlo simulations. Using the bond fluctuation model on a simple cubic lattice in the semi grand canonical ensemble, we locate the critical temperature of demixing via finite size scaling methods for a wide range of chain lengths (16 <= N <= 256 effective monomers per chain) and film thicknesses (2 <= D <= 19 lattice spacings). Simultaneously, we investigate the geometrical structure of the chains, showing that despite using melt densities there are pronounced "correlation hole effects", in particular for the smaller values of D. Also the components of the radius of gyration and end-to-end distance parallel and perpendicular to the confining walls are analyzed and their scaling behavior is studied. Evidence is presented that for strictly two-dimensional polymers (as occur for D = 2) the average number of intermolecular contacts scales with chain length N as z(c) proportional to N-3/8 and therefore the critical temperature scales as T-c proportional to N-5/8, whereas for values of D that exceed the excluded volume screening length, zc remains nonzero for N -> infinity, and hence T-c proportional to N. However, strong deviations from the Flory-Huggins theory occur as long as the unperturbed chain dimension exceeds D, and the critical behavior falls in the universality class of the two-dimensional Ising model for any finite value of D.