Solvothermal reactions of 2,5-thiophenedicarboxylate (H2tdc) and N-auxiliary ligands in the presence of cobalt(II), zinc(II) and cadmium(II) salts have given rise to seven new metal-organic frameworks, namely, [Co2(tdc)2(bpp)2(H2O)·(H2O)]n (bpp = 1,3-bis(4-pyridyl)propane) (1), [Cd(tdc)(bpp)(H2O)·(H2O)2]n (2), [Zn2(tdc)2(bbi)2·(H2O)2]n (bbi = 1,1'-(1,4-butanediyl)bis(imidazole)) (3), [Co4(tdc)3(bimb)4(H2O)5·(tdc)·(DMF)·(H2O)2]n (bimb = 4,4'-bis(1-imidazolyl)biphenyl) (4), [Zn4(O)(tdc)3(bimb)4·(H2O)5.25·(CH3OH)]n (5), [Cd4(tdc)4(bimb)6·(H2O)2·(CH3OH)2·(DMF)4] (6) and [Co2(tdc)2(tib)2·(H2tdc)·(DMF)2] (tib = 1,3,5-tris(1-imidazolyl)benzene) (7), which exhibit structural diversity. Compounds 1-3 display 3D 2-fold interpenetrating 8-connected tsi net, 2D 4-connected sql net, and 3D 5-fold interpenetrating 4-connected dia net, respectively; while compound 4 features an extremely rare 3D 7-connected hxg-d cationic net and compound 5 adopts a 3D trinodal (4,4,6)-connected net with a new topology structure. For compounds 6 and 7, the topology in the structures were determined as 3D 3-fold interpenetrating 5-connected bnn net and 3D 2-fold interpenetrating binodal (3,5)-connected net, respectively. In the solid state, complexes 1, 4 and 5 exhibit nonlinear optical properties, and complex 5 has also been evaluated as a promising visible-light-driven photocatalyst for degradation of organic pollutants. The magnetic investigation of complexes 1 and 4 shows that there exists spin-canted antiferromagnetism for 1, whereas strong antiferromagnetic interaction and/or the depopulation of the higher energy Kramers doublets of the Co(II) centers are dominant in 4.