Let's tackle the domestication issue first. There are really two routes to domestication of wild species: the more common one has been where humans have captured and bred animals as a food (and other) source. Think: cows, chickens, pigs, goats, sheep, etc. The path to domestication for dogs and cats was different: in this case, a somewhat symbiotic relationship occurred as humans started banding together into larger settlements: wolves began to frequent trash pits, and eventually began to help hunt and guard human settlements in exchange for a steadier food supply. As humans adopted cubs and started breeding them together, the evidence suggests that artificial selection progressed rapidly to select for juvenile traits: enlarged heads, big eyes, floppy ears, and a genial nature.
As for cats, they enter the narrative as humans domesticated plant species to create agriculture. The fossil evidence shows that rodents moved in to take advantage of stored grains as a food source, and then cats moved in to take advantage of the concentration of rodents as a food supply. It was at this point that humans likely started intentionally cultivating cats by allowing them into their homes, providing supplementary foods, etc.
Those cats that moved into human settlements to eat rodents were not in fact the larger wild felines like lions and tigers. The modern housecat is instead descended from wildcats (Felis silvestris, the forest cat), which are relatively similar in size. This makes sense: if housecats became domesticated after they took advantage of a concentration of rodents drawn to human harvested grains, then they'd be descended from cats that ate rodents, not from predators that pursued larger prey.
As for why they are so adorable: this is an educated guess, but it is likely because they resemble babies. Large heads relative to body size and large eyes are natural features of juvenile humans, and we're all descended from parents who thought those features cute, and therefore did a superior job of caring for their kids. Ergo, we find those features cute, and respond to other animals (mostly juveniles) that have the same features.