<p dir="ltr">The two-dimensional electron gas (2DEG) in shallow InAs quantum wells, combined with epitaxial aluminum, is commonly used to study topological superconductivity. Key features include strong spin-orbit coupling, a high effective g-factor, and the ability to manage proximity-induced superconductivity. My thesis discusses two aspects of this unique material. In the first section, I report on the transport characteristics of shallow InGaAs/InAs/InGaAs quantum wells and evaluate the effect of modulation doping on these shallow InAs quantum well structures. We systematically investigate the magnetotransport properties in relation to doping density and spacer thickness. Optimized samples show peak mobilities exceeding 100,000 cm²/Vs at n_2DEG < 10¹²cm^-2 in gated Hall bar, marking the highest mobility observed in this type of heterostructure. Our findings suggest that the doping layer moves the electron wave function away from the surface, minimizing surface scattering and enhancing mobility. This mobility improvement does not compromise Rashba spin-orbit coupling or induced superconductivity. In the second section, motivated by a theoretical study by Peng et al., we explore tunneling spectroscopy measurements on DC current biased planar Josephson junctions made on an undoped hybrid epitaxial Al-InAs 2DEG heterostructure. We observe four tunneling conductance peaks in the spectroscopy that can be adjusted by DC current bias. Our analysis indicates that these results come from strong coupling between the tunneling probe and the superconducting leads, rather than from Floquet engineering. We also touch on potential improvements to the device's design and materials. This work lays the groundwork for further investigation of Floquet physics in planar Josephson junctions. This thesis ends with a discussion of other unusual physics that could be explored in these novel shallow InAs quantum wells coupled with epitaxial aluminum.</p>