Galaxy structural properties such as size, morphology, and surface brightness bear the imprint of galaxies' evolutionary histories, and so are related with other properties such as stellar mass, star formation rate, and emergent spectra. In this dissertation, I present three studies exploring such relationships. In the first, I investigated the relationships between 4000 Å break (D4000) strength, colors, stellar masses, and morphology in a sample of 352 galaxies at intermediate redshifts based on photometric and spectroscopic data from the Hubble Space Telescope (HST). I explored several diagrams such as UVJ color space combined with the D4000 strengths and the structural parameters of sample galaxies. The analysis shows that the presence of a bulge component is a necessary but not sufficient requirement for star formation quenching at intermediate redshifts. In the second study, I investigated the central 250 pc UV star formation intensity (SFI, star formation rate per unit area) of a sample of 40 Green Pea (GP) galaxies and 15 local Lyman Break Galaxy Analogs (LBAs) to understand the Lyα escape mechanisms and the associations with the SFI in Lyα-emitters (LAEs). I utilized the Cosmic Origins Spectrograph near-ultraviolet (COS/NUV) images from the HST. I found that the Lyα equivalent width (EW(Lyα)) and the Lyα escape fraction are positively correlated with the ratio of SFI to galaxy stellar mass. These correlations suggest the importance of the central SFI in Lyα photon escape. In the third study, I investigated the UV photometric properties of a sample of 40 GPs and the possible associations with Lyα escape mechanisms. I measured the UV-continuum size and luminosity of the sample galaxies by employing the COS/NUV images. The circularized half-light radius of GPs shows compact sizes and it further shows the statistically significant anti-correlations with EW(Lyα) and the Lyα escape fraction. The size comparison of GPs to those of high-redshift LAEs shows that their sizes are similar, once spatial resolution effects are properly considered. These results show that a compact size is crucial for escape of Lyα photons, and that Lyα emitters show constant characteristic size independent of their redshift. Therefore, the results presented in this dissertation emphasize the importance of galaxy structural properties in star formation quenching and in Lyα escape.