Development of novel methods of molecular cooling and progress towards a general technique for molecular trapping are reported. Buffer-gas cooling of laser-ablated molecules is used to produce 1012 molecules of lead monoxide PbO in the ground X1Σ(v’’=0) state at 4 K, opening a possibility for improved sensitivity of searches for the electric dipole moment of the electron. A new method of loading atoms and molecules into a buffer gas cell is developed, and is used to produce 1012 Rb atoms at 4.5 K and $10^12$ molecules of imidogen NH X3Σ-(v’’=0) at <6 K. A next-generation trapping apparatus combining a molecular-beam loaded buffer gas cell with an in-vacuum 4.5-Tesla magnetic trap is developed. A critical parameter in buffer-gas-based magnetic trapping, the Zeeman relaxation rate in molecule-helium collisions, is measured for collisions of calcium monofluoride CaF with He. Up to $5\times 10^13$ CaF X2Σ+(v’’=0) molecules are produced via laser ablation at temperature of 2 K in helium buffer gas; the Zeeman relaxation rate is found to be $7.7\times10^{-15}$ cm$^{3}$ s$^{-1}$ . The result is consistent with a recent theoretical prediction, which indicates that the theory may be successfully used to predict suitability of molecular candidates for buffer-gas based trapping experiments and supports the continued use of new theoretical approaches to the calculation of molecular collisional processes.