Microscopic models for heavy-fermion materials often assume a local, i.e., momentum-independent, hybridization between the conduction band and the local-moment f electrons. Motivated by recent experiments, we consider situations where this neglect of momentum dependence is inappropriate, namely, when the hybridization function has nodes in momentum space. We explore the thermodynamic and optical properties of the highly anisotropic heavy Fermi liquid, resulting from Kondo screening in a higher angular-momentum channel. The dichotomy in momentum space has interesting consequences: while, e.g., the low-temperature specific heat is dominated by heavy quasiparticles, the electrical conductivity at intermediate temperatures is carried by unhybridized light electrons. We then discuss aspects of the competition between Kondo effect and ordering phenomena induced by intermoment exchange. We propose that the strong momentum-space anisotropy plays a vital role in selecting competing phases. Explicit results are obtained for the interplay of unconventional hybridization with unconventional, magnetically mediated, superconductivity, utilizing variants of large- N mean-field theory. We make connections to recent experiments on CeCo In5 and other heavy-fermion materials.