Spherically-symmetric solutions are relevant in many areas of cosmology, from perturbations in the early Universe to growth of large scale structures in the later eras. In this thesis, we first focus on a comparison between the tetrad-based method and the widely used Lemaitre-Tolman-Bondi (LTB) model for spherically-symmetric systems. We demonstrate that the tetrad-based method does not suffer from the gauge freedoms inherent to the LTB model, naturally accommodates non-zero pressure and has a more transparent physical interpretation.

Next we apply the tetrad-based method to a generalised form of Swiss cheese' model, which consists of an interior spherical region surrounded by a spherical shell of vacuum that is embedded in an exterior background universe, and verify the validity of Birkhoff's theorem at both the metric and tetrad level. Using this model, we reconsider critically the original theoretical arguments underlying the so-called $R_h = ct$ cosmological model, which has recently received considerable attention. These considerations in turn illustrate the interesting behaviour of a number of horizons' in general cosmological models. We also consider the theoretical arguments presented by Melia for the `zero active mass' condition, which he claims is required by the Friedmann-Robertson-Walker spacetime. We demonstrate that this claim is false and results from a flaw in the logic of Melia's argument.

We then use the tetrad-based methodology for modelling a cosmic void, in particular for the void observed in the direction of Draco in the WISE-2MASS galaxy survey, and a corresponding cosmic microwave background (CMB) temperature decrement in the Planck data in the same direction. We find that the present-day density and velocity profiles of the void are not well constrained by the existing data, so that void models produced from the tetrad based and LTB approaches can differ substantially while remaining broadly consistent with the observations.

We next consider the effect of pressure on perturbations. We develop both an analytic and a numerical approach for solving the field equations for a fluid with a fixed equation of state. We find an exact analytic solution for linearised equations, which may be novel in form, and which can be used to select the appropriate growing modes that can be used as an initial condition for evolving clusters and voids. Applying this to radiation as an example, we find oscillatory behaviour which corresponds to the initial stages of what become baryon acoustic oscillations. We then develop a numerical method for solving the field equations, which we use to compare behaviour of radiation waves in the non-linear and linear regimes. We find that non-linear oscillations travel faster than linear waves, which is interestingly analogous to non-linear waves in ocean waves. We also examine perturbations of fluids with a negative equation of state parameter, $w$, and find that at certain scales and range of $w$, it can support the growth of structure.

Finally, we consider the effect of pressure on photon propagation. We derive analytic expressions for pressure using a spherical top-hat density model, and use these to calculate the effect of pressure on the photon's path and energy. We find that the effect of pressure is negligible for fluids at cosmological densities and that it is valid to ignore it when propagating a photon.