Want proof? hold you cell phone near your computer speakers. Turn up the volume. Then make a call. You will be able to hear out loud the GSM signal. Although the microwave frequency isn't in the audible range, the envelope of the radio bursts is, so you are hearing a buzz formed by the radio packets going over the air.
You will only here these bursts when using the older 2G technology. 3G/HSDPA/LTE still interferes with equipment, but the designers specifically took the speaker interference into account. The radio bursts of newer phones are spread out in frequency and time, so that even though they are there, you can't hear them.
Sure, cell phones will probably work on airplanes. The range of a cell tower is a maximum of 30 km. How fast is the plane traveling? About 885 km/h, which works out to 14.75 km/minute. Cell phones only take about 5-10s to arrange a handover to another cell during a call, so it's certainly possible.
The problem is that since you are so far off the ground, and because the airplane is made of metal (impervious to radio signals), your phone will have to use the maximum transmit power. Such powers could easily interfere with radio equipment.
There is a theory on the Internet that, while in the air, your cell phone can see many too many different base stations at once, and it can't handle it. In my tests (done without a SIM card, so that no transmission can occur) I have not had good results. As a protocol stack developer on the BlackBerry, I can make it go into a special mode where it shows all of the cell towers that it can see. What I see in commercial airlines is that it may be able to see one or two cells, but they will very quickly disappear. There won't even be enough time to register.
The real dealbreaker is that airlines tend to fly over open space most of the time. Cell towers cover only moderately populated areas, and most of the time, you won't be in range of a tower