We use three-dimensional nonideal magnetohydrodynamic nested grid simulations to explore a variety of modes of core collapse and disk formation when the core emerges from a magnetically-dominated background. Once a protostar is formed, the accretion phase is characterized by a small amount of angular momentum but a large amount of magnetic flux in the near-protostellar environment. The low angular momentum leads to a very small (or even nonexistent) disk and weak outflow, while the large magnetic flux can lead to an interchange instability that rapidly removes flux from the central region. The effective magnetic braking in the early collapse phase can even lead to a counter-rotating disk and outflow, in which the rotation direction of the disk and outflow is opposite to that of the infalling envelope. The solutions with a counter-rotating disk, tiny disk, or nonexistent disk (direct collapse) are unique outcomes that are realized in collapse from magnetically-dominated clouds.