Do management typical typically study Computer Science?

Unfortunately most people become a manager by bring a stellar independent contributor. People management and engineering are very different skills, I'm always impressed when I see someone make that jump smoothly.

I always wanted companies to hire people managers as its own career path. An engineer can be an excellent technical lead or architect, but it can feel like you started over once you're responsible for the employees, their growth, and their career path.

Yeah, it just sucks that you eventually have someone making significant people management decisions without the technical knowledge of what the consequences could end up being. This would be even worse if you had people manager hiring be completely decoupled. The US military works this way and I have to say it's not the best mode.

Typically yes actually, the director of engineering should always be an engineer. Of course, these are hardware companies so it would probably be some kind of hardware engineer.

Should.

Sure.