Monolayers and heterostructures of two-dimensional (2D) electronic materials with spin-orbit interactions offer the promise of observing many novel physical effects. While theoretical predictions of 2D layered materials based on density functional theory (DFT) are many, the DFT approach is limited to small simulation sizes (several nanometers), and thus inhomogeneous strain and boundary effects that are often observed experimentally cannot be simulated within a reasonable time. The aim of this thesis is (i) to study effects of strain on 2D materials beyond graphene using first-principles and tight-binding methods and (ii) to investigate the effects of cuts--"kirigami"-- on 2D materials using molecular dynamics and machine learning approach. The first half of this thesis focuses on the effects of strain on manipulating spin and valley degrees of freedom for two classes of 2D materials--monochalcogenide and lead chalcogenide monolayers--using DFT. A tight-binding (TB) approach is developed to describe the electronic changes in lead chalcogenide monolayers due to strains that often persist in real devices. The strain-dependent TB model allows one to establish a relationship between the Rashba field and the out-of-plane strain or electric polarization from a microscopic view, a connection that is not well understood in the ferroelectric Rashba materials. This framework connecting strain fields and electronic changes is important to overcome the size and computational limitations associated with DFT. The second part of the thesis focuses on defect engineering and design of 2D materials via the "kirigami" technique of introducing different patterns of cuts. A machine learning (ML) approach is presented to provide physical insights and an effective model to describe the physical system. We demonstrate that a machine learning model based on a convolutional neural network is able to find the optimal design from a training data set that is much smaller than the design space.