Intermolecular Forces of Attraction
Assuming that by air bubble you're referring to the pocket of air now sealed in the bottle above the water surface:
Immediately after sealing the bottle, the air pocket is not at all compressed: it's at the same pressure as the surrounding environment (the air outside the bottle).

However, because you have a closed container, some water will evaporate, until an equilibrium is reached between the water vapor in the air pocket and the water in the liquid phase. This occurs when the chemical potential of the water in the two phases are equal.

Due to that vaporization, the total pressure in the air pocket (which now contains a higher amount of water vapor than the surrounding atmosphere outside the bottle) will be larger than the surroundings. This is a prominent effect when it's hot outside (have you ever left a sealed water bottle in a hot car?). 

While this effect is usually small, as gases mix very well, it is non-zero and so yes, the air pocket is slightly compressed due to presence of water vapor.

As for whether the air pocket can mix with water: Assuming our container can tolerate such pressures and we were to externally compress it: liquids are much less compressible than gases, and so the air would (given high enough pressure) condensate, mixing into the liquid phase (but you wouldn't achieve this by hand!)

To address your final point: let's say we invert the bottle, so the air pocket travels as a bubble through the liquid phase to re-emerge at the other end of the bottle. One could ask, "why doesn't the air bubble fragment into little bubbles as it progresses upwards?"

To be fair, you do see a few bubbles splinter off, but for the most part, the bubble is intact. The air bubble stays largely intact due to the viscosity of water (the intermolecular attractions between water molecules is larger than the entropic benefits of mixing with the air bubble). So, to answer your question, the fact that the air bubble doesn't mix is not due to its intermolecular attraction, but rather it's due to water's intermolecular attraction.