Noticed that bubbles could simply not be burst while underwater -- a bubble that was underwater with part of the surface touching the wall of the tub if pressed upon with a finger would simply move -- it would not burst underwater. What was I seeing?
A bubble in the air is a volume of air enclosed by a thin film of liquid.
A drop in the air is a volume of water surrounded by air.
The bubbles that you are talking about are volumes of air surrounded by water.
Their behavior would be analogous to drops in air, not bubbles in air.
Once a bubble forms in water, it means that the pressure of air in the water equals and opposes the pressure of water on the air bubble. This is a stable state unless there is extra energy involved. The bubble will go up due to buoyancy , so it acquires kinetic energy .
It could disintegrate to more bubbles if it scatters . Otherwise " diffuse into the water" is a very slow process, as the water is saturated with what air can diffuse in it
"or break into two or more smaller bubbles?" this can happen only through scattering, i.e exchange of energy. If the water is clear and there are no other bubbles above it , it will be stable until it reaches the surface.
if pressed upon with a finger would simply move
pressing with a finger introduces new variables, like water affinity to finger, and impulse. If enough energy and momentum is given , depending on the bonds of the water molecules, and the bonds of adhesion to the wall, the bubble would break to more pieces, think of the foam on the beach.