Is it possible to have switched and unswitched power in a circuit?

Inside the same device, you may have standby power, for remote control of the device. This standby power would be unswitched. The power switch or remote activation can throw the switch to activate the remaining circuits, to power on the screen, and audio circuits.

In a computer, power-on-lan or mouse, keyboard can bring a computer from sleep or hibernation to activate all circuits that were shutdown by timeout "power saving" features.

Therefore, yes, it is appropriate for some devices to have both switched and unswitched power in the same circuit board.

Within devices, not all circuits may be "active" at a particular point in the operation cycle of a device, where internal switches may become dormant or active. These circuits are often managed by gates, relays and sometimes actual switches, but they all perform the same function, to disconnect a portion from electric power at various times.

Sure it is, that is the function of a transistor. Many types of circuits have switched and unswitched power, commonly motor control circuits, the computer you are typing on has a motherboard that is switched and unswithched. (Oh you may be using a phone or a tablet those also)

Edit, it sounds like you are talking about industrial line machines here. If for instance you have a package line and the box machine goes down. The conveyor which is normally switched by the operation of the box machine still runs to clear the conveyor for maintenance. Like I say, motor controls are a frequent use of this