How fast does water cool off after boiling?

No one seemed that interested in answering the part of question about cooling time. Luckily, someone else has done a wee experiment and put it on their site:

http://jwilson.coe.uga.edu/EMT668/EMAT6680.2002.Fall/Ledford/ledford12/cooled%20_data.html

They also used a cooking pan.

According to their data and this excellent answer on Coffee SO which says:

Coffee solubles dissolve best at an optimal temperature of 195-205°F

the ideal moment to pour the coffee after boiling is around the 2nd minute, roughly between 90 and 150 seconds (though I've come to prefer cold brew, it tastes great and it's easy, which could be the real solution to your problem).

I'd wait about 30 seconds. If that doesn't do the trick or tastes burnt, try more coffee. If neither of those work, you may have a lighter blend of coffee (can also happen if it's old). Also make sure to steep for about 4 minutes on average for a French press before pressing, you can play around with times to find one that suites you best of course, but 4 minutes is the average amount of time it takes water at approx. 195 degrees f to extract the intended amount of flavor from of the coffee that the manufacturers shoot for without making "too strong" or "too weak". All subjective though. Hope this helps a little! Saw everyone else trying to be Isaac Newton and not trying answer your question so I figured I'd at least offer what I know. Cheers!

Boiling fresh water is indeed 100c or 212f at sea level. However your question is a very good one. If all the water in your pot were boiling, it would all vaporize very quickly.

The water just above the hot spots inside the tea pot are just under boiling, and reach boiling just as they vaporize, but the water convecting around the edges is much cooler.

In general, it is usually considered that if you bring tea water just to boiling point (the whistle just starts blowing), and you pour a cup of tea that water temperature overall is closer to 180F or 82C which is the perfect temperature for steeping tea leaves.

Coffee steeps faster in hot water as well. In fact you can make an appreciably better cup of coffee starting with hot water rather than cold water in a basic drip coffee maker. The hot water runs through the system faster which means less of the harsher elements of the coffee is extracted from the bean. Hot water equals swifter extraction which equates to better quality, where as just like with tea, If you steep too long, the tea loses its fresh quality and becomes astringent.

Tip: in your current phrasing, your question seems like a rather abstract physics question. You could get more informative answers if you expanded it to let us know what you are trying to prepare at a certain water temperature. Is it tea perhaps?

But to try to answer the first part of your question as stated: the Rouxbe cooking school has a video lesson demonstrating how you can identify different water temperatures without using a thermometer. For example, for the poaching cooking method (which is done in water at 71 to 85 degrees Celcius) you should look for the first small bubbles at the bottom of the pot and the first signs of steam from the surface. So assuming that the water is at 100 degrees Celsius as soon as you see steam forming is not necessarily correct. If you heat up the water further than the poaching temperature range, you get at the temperatures for simmering and gentle boiling. For a vigorous boil (100 degrees Celcius, which is the maximum temperature that water can reach at sea level) you have to wait until the water is moving and steaming faster, with big bubbles appearing on the surface.

The cooling rate will also depend on the mass (volume) of the water, the mass of the pot, the thermal transfer capacity of the pot and anything it contacts, ambient temperature, air pressure, humidity, purity of the water, etc. The answer to your question is "close enough".

As long as you are talking about a normal pot with or without a normal lid (i.e. not a pressure cooker) and you are reasonably close to sea level, you're right, boiling water is at 100°C. However, if you start to climb in altitude, that is no longer the case, at 300m, water boils at 99°C, at 600m, 98°C and so on. Wikipedia has a page with information about High altitude cooking that contains a reference table.