Your 'experiment' would not prove anything.
Yes, it does. Use my 30% example.

The theory of gravity
Exactly. You cannot prove it. It is not a law or a fact, it is a theory.
I can prove buoyancy, density and the electro-static force though, which yields the same effects as gravity and you can too. Science.

radio can get around the Earth to some extent by bouncing off the ionosphere
Yes, the "ionosphere" is the physical barrier above our heads and yes I do believe that they can bounce low-frequency signals off of it.

tides ... based on the motion of the Moon
False. The moon is not something that you can land on. It does not have mass that causes a "grav-itational" pull that overpowers earth "gravity". If it could pull an ocean (but not a building?) off the earth, then only one side of the globe would have a high tide at a time, not two. Science.

huge battery-powered clock that we live on
Yep, I made that term up because that is how I perceive it based on provable science. :)