dampening trainer sound

Turn your stereo up so he can't hear the trainer, problem solved.

Actually I would try the rubber mat trick the other poster mentioned and see if that worked.  Does it seem loud to you, some trainers are louder than others.

It depends on how the house is constructed.  If it's wood, you're screwed.  Unless you sound proof it like a music studio, the vibration would carry no matter what.  I used to ride the trainer in my exBF's townhouse living room (hardwood floor) so that I can watch TV.  The vibration shook the the entire unit plus the neighbor's.  Got a complain.  Then I moved the set up to the basement (concrete floor) which took care of the business.  A thick rubber mat would help dampen the vibration to an extent. 

I live in an apt building with hardwood floor, but the true floor is concrete.  I asked my neighbor and no one even notice I was riding. 

I would do what you could.  Yoga mat or rubber mats from the hardware store.  But I wouldn't go broke over it.  Maybe an extra piece of carpet.

I would tell them you have tried to do eveything you can to mitigate, maybe avoid any odd hours, but that's about all you can do.

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