Since at higher altitudes, the air pressure is lower, the boiling point of water decreases, since it's easier for the energy insde the water to get free.
When A liquid starts to boil, you reach a critical point where the liquid loses a lot of heat, much more than when not boiling, thus requiring much more energy for the same increase in temperature, and lowering the equilibrium point where the flame cannot increase further the temperature of the liquid.
An egg cook because the heat transform the proteins inside, tangling them with each other. But since the boiling water has a lower temperature, this process is slightly slower.
There are too many unknowns to model the situation with any accuracy.
If you know the power being pumped into your pan then you can easily calculate the amount of steam generated from the latent heat of vaporisation of water. You can calculate the power being generated by your cooker from the flow rate of the gas, or the current if it's an electric cooker, but it's anyone's guess what percentage of this is lost to the environment and how much ends up in the pan.
For any significant flow rate of steam I would guess that turbulent mixing will ensure the steam temperature is roughly constant throughout the pan. The temperature will be whatever the boiling point of water is at the internal pressure. The steam flow rate through the hole in the pan is fairly straightforward to calculate from first principles, though since steam is so important industrially I'd guess some Googling will find tables and empirical equations for flow rate.
Response to comment:
Let the mass of water lost per second be $m$, then the power applied to the water in the pan is just:
$$ W = mL $$
where $L$ is the latent heat of vaporisation of water. This will be equal to the power generated by your cooker times some unknown factor less than unity to allow for heat loss to the environment.
Response to second comment:
The temperature of the water will be close to the boiling point because any water hotter than the boiling point turns to steam, and the latent heat required will cool the water again. The steam in the layer immediately above the water will be at the same temperature as the water because it's in thermal contact with it.
If the steam above the water is hotter than the water you have to ask what is heating it. The only things I can think of that could heat the steam are the pan walls and lid. However the pan is only being heated from the bottom, and heat flow by conduction though the pan walls is a lot slower than heating/cooling by convection between the pan walls and the water in the pan. Therefore I would guess that the pan walls and lid are also close to the boiling point of water - actually they will probably be slightly cooler because they will lose heat to the surrounding air.
So I would guess that as long as there is enough water in the pan the steam in the pan will be close to the temperature of the water.
Best Answer
No. Boiling itself doesn't mean that the water will cook anything. If you have boiling water at 30°C you could touch it (if we forget that it's at really low pressure) and nothing would happen. Boiling is not what cooks, but temperature.
In fact, if you want to purify water at high altitudes, you need to boil water for a longer time because it will be at a lower temperature.