I investigate the role of disorder and its impact on the properties of a novel quantum crystal: solid $^4$He. The role of a superfluid field associated with edge dislocations on the properties of $^4$He crystal is studied at different levels of coarse graining. Initially, a study of the hydrodynamics of compressible superfluids in confined geometries as a coarse grained representation of superfluidity confined to complex networks is presented. The corrections due to finite compressibility to superfluid flow behavior are, as expected, negligible for liquid He. They are important but amenable to the perturbative treatment for typical ultracold atomic systems. Next, a study of the equilibrium properties of an Ising model on a disordered random network with quenched or annealed disorder is presented. We consider the transition temperature and other equilibrium thermodynamic properties, including those associated with one dimensional fluctuations arising from the chains. The transition temperature and the entropy associated with one dimensional fluctuations are always higher for quenched disorder than in the annealed case. These differences increase with the strength of the disorder up to a saturating value. The effect of the superfluid field on dislocation motion as a result of stress applied on the crystal is also studied. Damping of the dislocation motion, calculated in the presence of the superfluid field, is related to the shear modulus of the crystal. As the temperature increases, we find that a sharp drop in the shear modulus will occur when the superfluid field disappears. We relate the drop in shear modulus arising from the temperature dependence of the damping due to the superfluid field, to the experimental observation of the same phenomena in solid $^4$He and find good agreement. The response of the superfluid field to dislocation motion is studied within the quantum Gross-Pitaevskii formalism. The Dissipative Gross-Pitaevskii equation is used to investigate the effect of dislocation climb and glide motion on the superfluid field near it. Asymmetry introduced in the superfluid distribution due to dislocation climb is quantified. Unlike climb, glide motion does not affect the asymmetry characteristic of the superfluid distribution in the vicinity of an edge dislocation.