Conventional colloidal quantum dots (QDs) suffer from rapid energy losses by nonradiative (Auger) processes, leading to sub-ns lifetimes in all excited states but the lowest-energy single exciton. Suppression of interband Auger decay, such as biexciton Auger recombination, has been achieved with the design of heterostructured core-shell QDs. Auger-like processes are also believed to be responsible for rapid intraband hot-electron cooling in QDs. However, the simultaneous effect of shell growth on interband Auger recombination and intraband hot-electron cooling has not been addressed. Here we investigate how the growth of a CdS shell affects these two relaxation processes in CdSe/CdS core-shell QDs. Using a combination of ultrafast pump-push-probe spectroscopy on the QD ensemble and analysis of the photon statistics from single QDs, we find that Auger losses in the biexciton state are suppressed with increasing shell thickness, while hot-electron cooling remains unaffected. Calculations conducted within an eight-band k·p model confirm the experimental dependence of the biexciton Auger decay on the shell thickness, and provide insights into the factors determining the cooling rate of hot carriers.