Yes you can put the H-bridge in braking mode but then it acts as a damper. More resistance for higher speed. At zero speed the braking effect is zero.
Since you already have the thing dismantled, you can test braking for yourself. Spin the wheel with the motor disconnected. Now connect the two wires on the motor to each other. (An alligator-clip test lead is useful here.) Spin again. You will notice significant resistance. This works with any DC motor, including brushless ones.
For a centering force like a spring, you just need to read the position and then apply an analogWrite() value proportional to the distance from center.
It would be reasonable to assume that, with that power supply, the motor will be good with 24V and just a little less than 0.75A. That's not 100% guaranteed but it's unlikely to be using any fancy switching power supply to run the motor at a lower voltage.