We solve the Hamiltonian and momentum constraints of general relativity for two black holes with nearly extremal spins and relativistic boosts in the puncture formalism. We use a non-conformallyflat ansatz with an attenuated superposition of two Lorentz-boosted, conformally Kerr or conformally Schwarzschild 3-metrics and their corresponding extrinsic curvatures. We compare evolutions of these data with the standard Bowen-York conformally flat ansatz (technically limited to intrinsic spins χ = S/M 2 ADM = 0.928 and boosts P/MADM = 0.897), finding, typically, an order of magnitude smaller burst of spurious radiation and agreement with inspiral and merger. As a first case study, we evolve two equal-mass black holes from rest with an initial separation of d = 12M and spins χi = Si/m 2 i = 0.99, compute the waveforms produced by the collision, the energy and angular momentum radiated, and the recoil of the final remnant black hole. We find that the black-hole trajectories curve at close separations, leading to the radiation of angular momentum. We also study orbiting nonspinning and moderate-spin black-hole binaries and compare these with standard Bowen-York data. We find a substantial reduction in the nonphysical initial burst of radiation which leads to cleaner waveforms. Finally, we study the case of orbiting binary black-hole systems with spin magnitude χi = 0.95 in an aligned configuration and compare waveform and final remnant results with those of the SXS Collaboration [1], finding excellent agreement. This represents the first moving punctures evolution of orbiting and spinning black holes exceeding the Bowen-York limit. Finally, we study different choices of the initial lapse and lapse evolution equation in the moving punctures approach to improve the accuracy and efficiency of the simulations.