Type-II heterostructures are key elementary components in optoelectronic, photovoltaic, and quantum devices. The staggered band alignment of materials leads to the stabilization of indirect excitons (IXs), i.e., correlated electron-hole pairs experiencing spatial separation with novel properties, boosting optical gain and promoting strategies for the design of information storage, charge separation, or qubit manipulation devices. Planar colloidal CdSe/CdTe core-crown type-II nested structures, grown as nanoplatelets (NPLs), are the focus of the present work. By combining low temperature single NPL measurements and electronic structure calculations, we gain insights into the mechanisms impacting the emission properties. We are able to probe the sensitivity of the elementary excitations (IXs, trions) with respect to the appropriate structural parameter (core size). Neutral IXs, with binding energies reaching SO meV, are shown to dominate the highly structured single NPL emission. The large broadening linewidth that persists at the single NPL level clearly results from strong exciton-LO phonon coupling (E-ph = 21 meV) whose strength is poorly influenced by trapped charges. The spectral jumps (approximate to 10 meV) in the photoluminescence recorded as a function of time are explained by the fluctuations in the IX electrostatic environment considering fractional variations (approximate to 0.2 e) of the noncompensated charge defects.