We present 3D simulations of supernova (SN) explosions of nonrotating stars, triggered by the neutrino-heating mechanism with a suitable choice of the core-neutrino luminosity. Our results show that asymmetric mass ejection caused by hydrodynamic instabilities can accelerate the neutron star (NS) up to recoil velocities of more than 700 km/s by the gravitational tug-boat mechanism, which is enough to explain most observed pulsar velocities. The associated NS spin periods are about 100 ms to 8 s without any correlation between spin and kick magnitudes or directions. This suggests that faster spins and a possible spin-kick alignment might require angular momentum in the progenitor core prior to collapse. Our simulations for the first time demonstrate a clear correlation between the size of the NS kick and anisotropic ejection of heavy elements created by explosive burning behind the shock. In the case of large NS kicks the explosion is significantly stronger opposite to the kick vector. Therefore the bulk of the Fe-group elements, in particular nickel, is ejected mostly in large clumps against the kick direction. This contrasts with the case of low recoil velocity, where the Ni-rich lumps are more isotropically distributed. Intermediate-mass nuclei heavier than Si (like Ca and Ti) also exhibit a significant enhancement in the hemisphere opposite to the direction of fast NS motion, while the distribution of C, O, and Ne is not affected, and that of Mg only marginally. Mapping the spatial distribution of the heavy elements in SN remnants with identified pulsar motion may offer an important diagnostic test of the kick mechanism. Different from kick scenarios based on anisotropic neutrino emission, our hydrodynamical acceleration model predicts enhanced ejection of Fe-group elements and of their nuclear precursors in the direction opposite to the NS recoil. (abridged)
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