Like I said before, I was not required to have insurance as an employee.
Apparently different businesses do things differently. Based on my own experience, requiring an employee to have liability insurance is not the standard. I guess it depends on how big your business is, and the area of the country, what the standard is. There are probably different laws for different states, and perhaps if your business has loans, the bank may require different types of insurance.
None of the nurses I know have- or have to have, liability insurance. But, they are all employees. Employers have a huge responsibility for the actions of their employees. Also, it is generally the employer that has money (to be sued for) rather than the employee. Someone believeing they have a case for something, is not going to (or find an attorney to) to only sue Joe Blow nurse (who administered the wrong meds) who has no assets. They are going to sue the employer, who should have the right checks and balances, systems, oversights, training, so error does not happen.
Doctors tend to have their own practices. I don't know about doctors that work 100% as an employee of a hospital. A total guess on my part would be that some portion of their malpractice insurance is covered by their employer, and factored into their wages somehow.
Interesting questions.