Yes, you can connect a PC fan directly to the motherboard.
Yes, you can connect a PC fan directly to the motherboard.
Many options yes, though some use Molex connectors which you can find in product photos or specs. Look at the images or details to confirm the fan type you need. This model has a 4-pin header, allowing both PWM and pure voltage control. 3-pin connectors only support pure voltage control.
Yes, the CPU fan header and case fan header match exactly. The header I mentioned above the CPU is the CPU fan header, while the port for the CPU fan is now the case fan header. Those are the only two ports for a case fan. You can also use a MOLEX fan that connects to the power supply, but there are also more options like off fans. Keep that in mind.
Identify the specific kind of connector relevant to your needs.
There are three sections for fans listed. The one you might have overlooked is near the bottom left of the CPU socket.
If you receive an adapter, that's fine. But fundamentally, it's not.