Never seen printer with a heated bed and a power supply only rated for 20 amps. Well, unless the HIC powers the heat bed with 120V line voltage or something. If you are printing on a warm/hot bed, try running a print with the heat bed turned off. Like 3dkarma suggested, maybe the power supply is dropping out when the motors are added into the mix. If you have a voltmeter, monitor the 12V output of the power supply and see how stable it is. If the power supply is bogging, it could be affecting the motors too. Also, if the y-axis is belt driven, make sure the belt is tight so that it can't slip on the motor pulley.

I've also never, ever seen a video of a hot end dripping melted drops of filament like that. That suggests to me the hot end is actually a LOT hotter than you think it is. The extruder is supposed to soften the filament, not turn it into a liquid state. Usually when a hot end drifts over time, you end up with a spaghetti pile of filament or what people describe as a nest. Maybe a dumb question, but is there a chance that the connections for the hot end and heated bed temp monitoring thermistors are swapped at the electronics?