1. How many people work in the several hundred government departments and agencies
of the executive branch?

Around 2.7 people if we exclude the military personnel, Around 4 million people if we include them

Explanation:

Executive branch has the role to implement the laws and programs created by the government. Since they have to do this nation wide, this branch require a large amount of employees.

For comparison, both judicial and legislative branch only employs around 30,000 employees. Basically  Almost all government officials that get in touch with the people are part of the executive branch. Fire fighters, police and social workers are all parts of the executive branch.