Pure Blood devil and their Chess pieces
I read the light novel, and you got it wrong. It wasn't that only 3 devils survived. Think of it more like they were an endangered species (more than 3), and to increase their number one devil created the chess pieces. When a devil wants a servant, it doesn't limit to just humans, it can be any creature (devils included, i.e. Ravel). For example, a dragon that trains Issei is actually a devil as well. As for someone having to die before they can be a devil is false, it's more like they are just reborn as a devil when they are turn into a servant.

Both Rias and Ruval are pure bloods meaning their parents are devils. In Rias' case, she has no King, but she could if another devil wanted her as their servant (which is the case with Ravel).

Lastly, not all Kings are pure bloods. I'm pretty sure the first season explains that if Issei tries hard enough, he can get his own set of chess pieces making him the King.