There are two unrelated effects at work here. One is the atmospheric pressure, and the other is the surface tension of the water.
Start with you holding the plate in place, and consider what happens when you release the plate. For the plate to fall down one of two things must happen. Either the volume of the water in the glass must increase, to allow the plate to move down, or air must flow into the glass at the contact line between the glass and the plate.
Consider the first of these. If you pull the plate down slightly (and no air leaks in) the volume inside the glass must increase. Water has such a high bulk modulus that we can approximate it as incompressible. You would need an immense force pulling down on the plate to stretch the water to any significant degree. In practice the water would boil before its volume increased significantly, but even the lesser force required to boil the water is far greater than the weight of the glass plate.
So the only way the plate can move down is for air to leak in through the contact between the glass and the plate. However this means forming small bubbles at the contact line, and small bubbles have a very high pressure due to the surface tension at the air/water interface. This means the bubble formation requires a greater pressure than the weight of the plate can generate, so the plate can't move down this way either.
Incidentally, the effect of surface tension explains why the plate won't stick if the glass/plate contact isn't very good, or if there's a chip in the glass. In both cases there is a relatively large gap where a bubble can form, and large bubbles have a smaller pressure than small bubbles (the bubble pressure is inversely proportional to the bubble radius). The weight of the plate can generate enough pressure to form the large bubbles, and the plate falls off.
Now we can explain why the plate falls off when you immerse the glass and plate in water. If you do this there is no air/water interface at the contact between the glass and the plate, so there is no surface tension effect. Water can leak through even the tiniest gap between the rim of the glass and the plate so the plate falls off (though it may take a few seconds as the water won't flow in instantly).
The trick will work with most liquids because most liquids will neither expand nor boil under the weight of the plate. However it wouldn't work with very volatile liquids like ether, because ether boils too easily and vapour bubbles will form in the glass. You'd probably also find it wouldn't work if the air/liquid interface has too low a surface tension, because a low surface tension allows air to lean in between the glass and the plate.
Some of your requirements are conflicting. The surface pressure of any atmosphere is the weight of the column of air (whatever mixture of gasses your atmosphere is made up of) of unit area extending upwards to the end of the atmosphere.
The surface pressure is therefore a function of the gravity and how much total gas is in the atmosphere. For example, Earth and Venus have about the same gravity, but the surface pressure of Venus is about 90x more because Venus has about 90x more stuff in its atmosphere.
You can generally assume that any planetary atmospheric layer will be thin enough that gravity is constant over that layer to a reasonable approximation. Therefore, for any chosen gravity, how fast the pressure decreases only has to do with the density of the gas. The denser it is, the faster you have a smaller weight of air column above you as you go up in altitude. To have a planet with 1 g surface gravity, for example, that has a more rapid falloff in pressure than earth, you need its atmosphere to be made of a more dense gas. The denser the gas, the thinner the atmospheric layer for the same gravity and surface pressure.
This is why the larger planet in your description just can't be. You say it has less gravity than Earth, a thinner atmosphere (presumably you mean lower surface pressure), but yet you want the pressure to fall off more rapidly than on Earth. This simply can't happen. Picture one inch square on the ground and the colum of air above it to the end of the atmosphere. With less gravity it will be less squished, so the column will be higher. This in turn makes the pressure fall off more slowly with altitude. If you also say it is a nitrogen-oxygen mixture, then it's density is pretty much defined over a narrow range (N2 has a molecular weight of 28 and O2 of 32).
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At any point above or below the Earth's surface the atmospheric pressure is equal to the weight of air above a square metre surface. So if you go down a mine the atmospheric pressure goes up because there is more air above you.
If the density of the air varies with height as some function of height as $\rho(r)$ then the pressure at a height $h_0$ will be given by:
$$ P(h_0) = \int_{h_0}^\infty \rho(h)g(h)dh $$
where $g(h)$ is the gravitational acceleration. Although this looks simple enough the density of the air varies in a complicated way because it's affected by temperature differences. As you down e.g. a mine the temperature goes up and the air density goes down. I found some calculations of the pressure change in this paper, though the authors bemoan the fact that no actual experimental measurements exist in the literature.
Re your other questions, I believe the upper pressure limit is due to the toxicity of nitrogen at high pressures rather than your ability to breath. The proportion of oxygen in the air wouldn't change (much) and since the density is increasing the amount of oxygen per cubic metre of air would indeed increase with depth. I doubt pressure changes would be detectable on your skin, though most of us have detected pressure changes when our ears pop.