Does the required torque to spin a propeller increase with RPM? I‘ve currently been experimenting with different propellers: 12x12 prop, 16x12 prop (both two blades) and a 22x14 (6 blades) prop. They were propelled by a 2.4 kW electric motor with 260 kV on a 6S/5200mAh/12C battery setup.

The 12x12 prop produced 2 kg of thrust at full throttle, the 16x12 produced 4.5 kg of thrust. So I thought the 22x14 would produce around 6 kg because its bigger blade area would be more efficient. However, I got only 0.8 kg of thrust and recognized that the motor would slow down at a certain RPM.

I have to say the big prop is pretty heavy so could it be that the motor would have enough power to spin the prop faster but it lacks enough torque to get it there?
Of course. More and bigger blades is more drag. Drag is also proportional to the velocity squared. Even maintaining the RPM, a bigger propeller (diameter) will have the additional diameter flying faster.

To maintain a higher RPM (or to use a bigger prop) the force to cancel the increased drag is higher, and thus the torque (force times distance) that drives the propeller.

Place the propeller in vacuum, and the engine will have no trouble with the propeller apart from any loads on the shaft (but there won't be any thrust).