We search for the flavor-changing neutral-current decays B -> K-(*()) v (v) over bar, and the invisible decays J/psi -> v (v) over bar and psi(2S) -> v (v) over bar via B -> K-(*())J/psi and B -> K-(*()) psi(2S), respectively, using a data sample of 471 x 10(6) B (B) over bar pairs collected by the BABAR experiment. We fully reconstruct the hadronBic decay of one of the B mesons in the Y(4S) -> B (B) over bar decay, and search for the B -> K-(*()) v (v) over bar decay in the rest of the event. We observe no significant excess of signal decays over background and report branching fraction upper limits of B(B+ -> K+ v (v) over bar) < 3.7 x 10(-5), B(B-0 -> K-0 v<(v)over bar>) < 8.1 x 10(-5), B(B+ -> K*(+) v<(v)over bar>) < 11.6 x 10(-5), B(B-0 -> K*(0) v<(v)over bar>), < 9.3 x 10(-5), and combined upper limits of B(B -> K v<(v)over bar>) < 3.2 x 10(-5) and B(B -> K* v<(v)over bar>) < 7.9 x 10(-5), all at the 90% confidence level. For the invisible quarkonium decays, we report branching fraction upper limits of B(J/psi -> v<(v)over bar>) < 3.9 x 10(-3) and B(psi(2S) -> v<(v)over bar> < 15.5 x 10(-3) at the 90% confidence level. Using the improved kinematic resolution achieved from hadronic reconstruction, we also provide partial branching fraction limits for the B -> K-(*()) v<(v)over bar> decays over the full kinematic spectrum.