Sounds like you have an intermittent switch contact on your controls. This could cause both the hum and the motor stoppping while the tape is playing. If you feel up to it, try this: Get yourself a can of De-Oxit (You can order cans from Amazon). Take the tape player out of its case (carefully!!!). Find the switch(s) that control play and give them a LIGHT squirt of De-Oxit and work the switches through their entire range. While you have the deck open, go ahead and treat any other controls likewise. Be VERY CAREFUL not to get any of the De-Oxit on the belts and other drive parts of this unit. Let the De-Oxit dry, run a clean cloth over the interior to clean any overspray and put the unit back together.
Grounding: All the components of your system should be grounded back to a common ground point, usually the ground lug on you amplifier. From the ground lug, you can then connect a lead to a good ground source, like a cold water pipe or other good ground. "Daisy Chaining", which is connecting one unit to another and to another, etc, is not recommended.
Hopefully, between the cleaning and grounding, you will have a quiet well performing system again. I am sure other forum members will also chime in with good recommendations.
By the way, welcome to the wonderful world of Quad.
MTGC (Michael)