2013 Nissan Rogue clicking behind dash on cold start

You're on the right track, that's the blend door actuator.

Every time you start the car, the HVAC system runs a quick calibration sweep of its actuators. Inside the actuator there's a small plastic gear train, and when those teeth strip, the motor keeps spinning and clicking against the dead gears instead of parking the door. That's the rapid clicking you hear behind the glovebox. It stops on its own once the system times out, which is why it lasts about 10 seconds.

To confirm it, sit in the car with the key on and the engine off so it's quiet. Turn the temp knob from full cold to full hot. If the clicking comes back or the vent temperature doesn't change, you've found it. You can run the same test switching vent modes to check whether the mode door actuator is involved too.

The clicking itself won't hurt anything. The real problem is that the door eventually stops responding to the temp knob, so you end up stuck on hot or stuck on cold.

The actuator sits on the HVAC box right behind the glovebox. On these Nissans you can usually reach it without pulling the dash, a couple of screws and one electrical connector. It's a straightforward driveway job if you're patient in the tight space. If the door shaft inside the box is cracked instead of the actuator gears, that's a bigger job, so check the model-specific service information before ordering anything.