Does solute concentration vary significantly with depth?
(1)If I dissolve glucose in an ordinary beaker filled with water, are the concentrations of glucose at the top and bottom significantly different?
(2) What if we use NaCl as the solute?

1 &amp; 2 are basically the same question. The difference in gravity between the top of the beaker and the bottom of the beaker is real but insignificant. So due to gravity alone there won't be a concentration gradient.
However there are solutions which can be centrifuged which will create a concentration gradient. Essentially the centrifuge creates a large gravity gradient. Close to center gives lower gravity, far from center gives higher gravity. Once you stop the centrifuge however the concentration gradient is not at equilibrium. So over time the concentration gradient would disappear.

(3) Is there a molecule large enough to ``sink'' to the bottom, and if so would it still be classified as a solute?

The above discussion about a centrifuge confirms the basic notation, but there is an unstable gradient, not a sinking per se.
Obviously if you drop a marble in a glass of water it sinks. But the marble isn't dissolved in the water.

(4) Likewise, are there solutes which are lighter than H2O, so they float upwards?

This is a bit more interesting. Open a plastic bottle of soda and put a balloon over the neck of the bottle.  Carbon dioxide was dissolved in the soda under pressure. When the soda is opened, the pressure is released and the excess carbon dioxide will &quot;float up&quot; out of the soda and blowup the balloon. Since the gas phase is less dense than the liquid phase (soda) it is on top. The gas doesn't really &quot;float&quot; on the soda however. It is also important to realize that once equilibrium is established the carbon dioxide remaining in the soda is homogeneously distributed throughout the soda.
A different sort of twist are soap molecules which will &quot;float&quot; to the surface, not because they are lighter, but because of the surface tension of water. The famous soap/pepper experiment illustrates this.
Another sort of twist here is a bartender who makes fancy cocktails by floating one layer of a liquor over another due to the higher layers being less dense. The segregation of the layers is unstable however. One good swirl with a straw and the layers mix.

For 3 and 4 there is what is called a phase diagram. For example it shows if two liquids will mix, or if there will be two phases where one liquid would float on the other.
Think of gasoline and water. There is a gasoline layer floating on the water so there are two liquid phases because there is an interface. However a very tiny amount of gasoline will be dissolved in the water and a very tiny amount of water will be dissolved in the gasoline. In this case the gasoline that is dissolved in the water is distributed homogeneously throughout the water. Likewise the water dissolved in the gasoline is distributed throughout the gasoline.

I used gasoline as a conceptual example. Gasoline however isn't a pure compound but rather a mixture. So for those that want to be picky substitute n-hexane for gasoline.