One of the requirements to getting a PhD is the knowledge that a PhD is not the proof of intelligence you thought it would be when you started. It is better a proof of resilience, patience, and being dead inside.
Agreed. In my experience, arithmetic skills are not transferable to higher-level math skills and vice versa. Of course, someone can be skilled at both but I don't think you automatically become good at one because of the other.
I also have a PhD in Computer Science and I assure you that the degree does not give me as much blind faith in my math ability