There's still something missing from all the answers so far. When you drop something on the ground, say, a rock of mass $m$, by the time it makes contact with the ground it's traveling at a velocity $v$ and thus has momentum $p = mv$. To be stopped completely, its momentum has to equal $0$ at the end. So you have a total change in momentum of $\Delta p$. According to (the most literal, I think) Newton's 2nd law, you have $\Delta p = F \Delta t$, where $F$ is the force slowing down the object over the timespan $\Delta t$ (in reality time is continuous and $F$ is probably changing continuously, but this is enough to illustrate the point).
So, if the $m = 1\ kg$ rock goes from falling at $v = 10\ m/s$ to $0$ in a millisecond or so, you might have $F = \Delta p/\Delta t = 10\ kg\ m/s /(.001s)=10000\ N$, which is obviously much bigger than just the gravitational force of $F_g \approx 1\ kg \times 10\ m/s^2 = 10\ N$.
Your confusion stems from not clearly defining your (two) systems.
- For the Weights:
There is one system in which you would analyze the dynamics of only weights in your hand, and then consider all the external forces acting upon it. In this scenario, there is gravitational force and the reaction force from your hands. That's it.
- For yourself plus the weights:
In this system, you will now consider your own dynamics and look at only the external forces acting upon yourself. As again, there is gravity! But, note that gravity does not see you or the weight in isolation. Hence, $F_G = g (m_{body} + m_{weights})$, is the gravitational force. And then you are still standing on ground, which means there is a reaction force.
So you see, that equation is for system 2, and your doubt is in system 1.
I think the key-point is that in rigid-body dynamics, you ignore internal forces (since the body is rigid) and look at the dynamics governed by all external forces.
Best Answer
Yes, at approximately the speed of sound.
Yes
The swimming one can be quantitatively proven with a kitchen scale. Weigh an object that will float. Fill a bowl partway with water. Weigh it without and then with the object.
The airplane one can be qualitatively seen in pictures of aircraft flying low over water.