UnitedHealthcare operates in DC and all fifty states and has north of 26 million customers. Kaiser operates in eight states and DC and has around 12 million customers. It's safe to say plenty of folks use Kaiser when it's available.

The downside is that because Kaiser operates as an HMO, any specialists you need to see must be approved by your primary care physician (typically GP, ob-gyn, or uro). They have an abysmal record with mental health, and folks who need chronic non-routine care often struggle to get that care. If all you need are routine checkups, cheap prescriptions, routine immunizations, etc. they're quite competent.

  Who specifically sets the prices?

Kaiser is a vertically integrated HMO. Like other HMOs, to get coverage (with a few exceptions) you need to see a provider within the (Kaiser) network. As a vertically integrated HMO the providers in the Kaiser network are all employed directly by Kaiser. So Kaiser sets the its own employee compensation.

It's worth noting that all insurance companies are required to emergency care at any provider as if it were in network. If a Kaiser member were to go to a non-Kaiser ER they would see the same cost as if they'd gone to a Kaiser ER. Whatever Kaiser pays to the out of network hospital/staff is almost certainly negotiated beforehand.