Every business has insurance, including insurance companies. This is because they specialize in their own business and outsource non-core business.

An insurance provider for a specific industry has teams of skilled attorneys that are expert in that specific domain. Cigna might self insure if they consider medical malpractice a core part of their business. In that case, your attorney sends the letter to their corporate counsel.

Often times, day to day business people just ignore liability risks because it's not in their core mandate. But the corporate counsel is in charge of managing legal risk, and they will see such a situation as unacceptable. Corporate counsels also hold a lot of sway within a businesses structure.

Yes, it's called reinsurance: https://en.wikipedia.org/wiki/Reinsurance

Yes, they typically will buy insurance to deal with risks outside their core business (like the health insurer having liability coverage).