It's difficult to understand your theory because it is very non-intuitive to me and because it is in stark contrast with my prior knowledge. Good enough?
Yep, good enough.
The above questions are I believe quite relevant. Pressure as we know it is always caused by something, and in our different balls example it seems to be caused by the objects the air surrounds; through what mechanism the density of an object determines what amount of air amasses by it's surface?
No mechanism as such. It's the object itself and the density of it which determines how much atmospheric pressure it displaces, which is what's calculated on a man made measuring scale.
Take away the pressure upon any mass and you take away the mass itself. If you were to take all of the pressure, I mean.
If you were to take away a portion of pressure on that mass, then the measured weight would decrease, because that mass is not resisting that pressure and neitehr are the scales.
Could we measure these pressure differences next to objects of different density?
Believe it or not, you already are. Every time you place any mass on a scale, you are not measuring the mass, you are measuring the resistance of that masses push against the atmospheric pressure above it by what it displaces due to it's density.
Is air a special case or do objects of different densities displace different amounts of any fluid they're surrounded with?
They displace different amounts, it's just that people don't realise it because most people simply accept gravity for things falling and assume that's that.
Here's an example, so think about is seriously.
The reason why a block of lead and the same size sponge weight differently is simply due to how much atmospheric pressure their density can displace.
We know that lead can displace much of it's full density against that pressure, as it absorbs very little into it.
The sponge although the same size; absorbs most of the air pressure into it, leaving very little of it to push back. to give an idea of what it's pushing back, just imagine compressing that sponge as tight as you possible can. You would end up with a pea sized sponge, right?
This is all the atmospheric pressure that the sponge is really displacing.
Equate that to scale measurement and you can see why lead is heavy and a spong is light.
See what I mean?
And from an earlier question that I think was not fully answered:
sceptimatic:
"Naturally it's not as strong a push as it is when the ball is acting against the higher pressure because it loses energy."
Energy. You established that the ball gains energy in form of compression and heat (in minuscule amounts), and it appears that the energy contained by a moving object is producing a force to keep it moving by spending this energy. Could you explain how unnoticeably small amounts of heat and compression do work to move our medicine ball upwards?
Ok, now think of this in super fast motion to get the picture.
Imagine you throw that ball up into the air at 1000 mph (for instance). If you slowed the camera down, you would see the front of the medicine ball compress a good bit. Basically it's expanded itself sideways or down its sides, creating a super fast friction around it due to the compressed top of the ball. This friction would heat up the ball, creating a lower pressure around and back under it which has to be equalised and it does by a squeeze of atmosphere against that lower pressure.
the best way I can describe it without confusing it, is by imagining the ball being a wet bar of soap in your hand and you are trying to grab it and squeezing it up every time you do. As long as your energy is applied like that, it will keep going.
As soon as you stop, then the friction against the compression up top, starts to exert that force back due to you no longer applying energy.
Now I admit, I've went extreme here to show you how it works but the same applies under a small energy load, it's just that we cannot see the effects with out puny throws.