All of the “paradoxes” you mentioned can be explained by the fact that the 4 fundamental forces simply have different relative strengths. At different distances, different forces are dominant. So you are roughly right in saying that “matter works differently at different scales.”
As @jammycakes pointed out, @T_aquaticus is wrong in saying that gravity decreases more quickly than the EM force. They are both inverse squared laws. But he is right in pointing out that gravity is much weaker than the EM force, if we look at the everyday objects around us and calculate how much gravity vs. EM force they would result in. James is also correct in pointing out that the EM force is very easy to cancel, as there are positive and negative charges all around us (in fact, in every atom).
In contrast, as far as we know there is no such thing as “anti-gravity” to cancel normal gravity. (Interestingly, some physicists at CERN are testing that right now by investigating if antimatter drops the opposite way from gravity.) Gravity is a very simple force: it exists between any two massive bodies. Einstein showed that you can understand it as the existence of mass distorting the fabric of spacetime. I’m not an expert on GR, but for me, the universality of gravity to me is what makes such a theory intuitively plausible. (It’s hard to imagine explaining the EM force as coming from distortions of spacetime.)
At large scales, gravity becomes the dominant force: besides the weakness of the other forces, there are large masses of matter in space whose gravitational attraction cannot be canceled out, such that the small value of the gravitational constant is made up for. If you pack enough matter in a small enough volume, the force will be enough to make the mass collapse on itself and form a black hole. However, it’s hard to imagine this happening with EM force, as positive charges attract negative ones, so you can never create a black hole by amassing tons of positive charges or negative charges.
(One can imagine, though, a universe where the gravitational constant is much, much larger, such that even small masses of things will collapse into a black hole. That would make planet formation impossible. So perhaps this is an example of fine-tuning (?))
I found this article by Matt Strassler to be very comprehensive and well-written about exactly this issue: