Topological quantum computation using non-Abelian Majorana zero modes localized in proximitized semiconductor nanowires requires careful electrostatic control of wire junctions so as to manipulate and braid the zero modes enabling anyonic fault-tolerant gate operations. We theoretically investigate the topological superconducting properties of such elementary wire junctions, the so-called T junctions, finding that the existence of the junction may nonperturbatively affect theMajorana behavior by introducing spurious nontopological subgap states mimicking zero modes. We propose a possible solution to this potentially serious problem by showing that junctions made lithographically from two-dimensional (2D) electron gas systems may manifest robust subgap topological properties without any spurious zero modes. We propose a 2D structure that enables multiprobe tunneling experiments providing position-dependent spectroscopy, which can decisively settle outstanding open questions related to the origin of the zero-bias conductance peaks observed experimentally. We also find that junctions with trivial superconductors may result in local perturbations that induce extrinsic low-energy states similar to those associated with wire junctions.