Please find the answers to your questions below:
First, very carefully balance the telescope and mount, both in RA and DEC axis. Stepper motors (and the drive train in general) will probably not be damaged by trying to operate out of balance, but the stepper motor will miss steps or the belt will slip and not move the mount as far as expected when the resistance to motion is too great. You said you can move the mount with slow motion controls. Is this easy to do, or are the knobs providing resistance when you try to turn them? It should turn with almost no resistance - like turning the volume knob up on a TV or radio, Any resistance to movement could mean that the mount needs to be cleaned and greased, or that your balance is off, or that your worm drive mechanism needs adjustment, or (if the resistance gets higher and over as you rotate the knob) the drive gears may be damaged.
Just to confirm, the smaller pulley (the 16 tooth one) is on the motor and the bigger one (40 tooth) is on the shaft connected to the worm gear on the mount. And the belt should be pulled tight on the pulleys so you cannot press it in or out in the center between the pulleys more than a mm or two. Any looser than that could allow it to slip over the teeth.
(Assuming you are in the northern hemisphere,) if you hold the SOUTH button for many seconds, does the mount swing the telescope to point below the horizon in the southeast or southwest? If instead it rotates around the pole then you have the wires for the DEC and RA motors connected to the wrong motors. I have done this, it is confusing when it happens because often the mount will move in sort of the right places but be off by a bit. It becomes obvious when you move one axis at a time, but when you do a slew to start your alignment it is hard to see.