The operator algebras generated by translation and modulation unitary operators can play an important role in studying Gabor frames in time-frequency analysis. We present some connections between these algebras and Gabor frames, and use them to derive some well-known results and their generalizations such as the density property in Gabor analysis, as well as some new ones such as a characterization of the Gabor frames (for subspaces) admitting unique Gabor duals (within the subspaces). Finally, we provide a necessary and sufficient condition for a square-integrable function $g$ to generate a subspace Gabor frame in the one-dimensional, rational case. The condition is phrased in terms of the Zak transform of $g$.