We consider generalizations of a sum, which was recently analyzed by Pemantle and Schneider using the computer software Sigma, and later also by Panholzer and Prodinger. Our generalizations include Tornheim's double series as a special case. We also consider alternating analogs of Tornheim's series. For Tornheim's double series and its alternating counterparts we provide short proofs for evaluation formulas, which recently appeared in the literature. We introduce finite Tornheim double sums and alternating analogs, and provide relations to finite multiple zeta functions, similarly to the infinite case. Besides, we discuss the evaluation of another double series, which also generalizes Tornheim's double series.