We do not have or know where to get a manual for the TaoTao ATE501 or its rear hub assembly. Looking at your photo, the new freewheel sprocket appears to be correctly installed and looks normal to us. The old freewheel sprocket most likely locked up from internal wear caused by old age.
Yes, that amount of gap between the freewheel sprocket and the frame's rear swing arm is normal for the Taotao ATE 501 electric scooter.
Matt wilson