To figure out why this happens, you need to think about what boiling is, and how it works.
As you would know, the water in the pot boils because its temperature was raised above the boiling point by the flame. This required a net transfer of heat from the flame, through the pot, to the water in the pot. Why did the heat flow in this direction? Because the flame is hotter than the water in the pot, even when the water starts boiling ($T_{flame} > T_{boil}$)
Now, think about the water in the bottle. The only way for it to get heat is through the water in the pot. As long as the temperature of the water in the pot, $T_{pot}$, is less than $T_{boil}$, it is still liquid, and it transfers some heat to the water in the bottle. The water in the pot boils off at $T_{boil}$, and can no longer transfer heat as efficiently to the water in the bottle.
This effectively means that the water in the bottle is restricted to a maximum temperature of slightly less than $T_{boil}$, and that is why it never boils.
Another way to think of this is, there must be a temperature difference for a heat transfer to take place. Since the maximum possible temperature of the pot water is $T_{boil}$, the temperature of the bottle water can never exceed this.
EDIT: Another factor to consider is the low conductivity of glass, which means a high temperature difference is required to let a small heat flux through.
Yes, your analysis is correct. Water in equilibrium with ice is at a temperature of 0 degrees C. The reason that the water doesn't spontaneously turn to ice has to do with the latent heat of fusion of water: in order for water to turn to ice at zero degrees C, you need to remove quite a lot of heat from it. In the case of water / ice, the latent heat is almost exactly 80 cal/g (334 J / g). Another way of thinking about this: one gram of ice at zero C can reduce the temperature of 1 gram of water from 80 C to zero C (or 4 gram from 20 to 0, etc). This is why you can add a few chunks of ice to a drink, and the whole drink will get cold without getting too diluted. Adding more ice than is needed helps speed up the process (because the surface area in contact with the liquid is larger) but you will end up with some solid ice at the end - and this in turn will keep the liquid at 0 C since any heat absorbed from the environment will quickly go into melting a bit more of the ice.
Best Answer
There is no easy way to calculate this for liquids because the heat exchange will depend on whether there is any convection in the liquid or not. You can calculate the solution for the heat (conduction) equation for your geometry, but this may or may not give the right answer. The problem is a lot better defined for solids which can not convect.
The solution to the heat equation in cylindrical coordinates can be found in many physics books and scripts. It will neglect the bottom. There is another question that you need to consider: how homogeneous does the temperature have to be in the bootle? Unless the liquid is being stirred, there will be a significant gradient. If you look at the way chemists are doing their temperature dependent reactions, there is almost always some rather strong stirring or agitation going on, otherwise things may react differently in one part of their beakers than in another. You may want to stir, too.
The good news is that stirring simplifies the physical problem and it will, at least that's my gut feeling, reduce the time for the heating and the variation in the time it takes to get to the right temperature. The heat equation can then be reduced to the boundary, which is characterized by an effective area and an effective thermal resistance.
As a final comment: I would never leave the health of my baby to theoretical calculations. The only safe method is to measure the temperature.