A transponder is not required unless an aircraft is operating: In Class A, Class B, or Class C airspace. Above 10,000 feet Mean Sea Level (MSL), excluding airspace below 2,500 feet Above Ground Level (AGL).
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The quick answer is NO, pilots cannot turn off the transponder while the aircraft is in flight. The transponder is a vital piece of equipment that is used to identify the aircraft to air traffic control. It also transmits important data about the aircraft, such as its altitude and speed.
However, if you wish to operate in class A, B, or C airspace, or at an altitude of over 10,000' MSL, or within a 30 nautical mile radius of the primary airport in class B airspace, you will need a transponder and altitude encoder (commonly referred to as "mode C").
— The device could have an electrical short or catch on fire. Pilots would want to shut it down rather than risk a fire spreading to the rest of the cockpit or airplane.
It's also worth noting that if the transponder is switched off you generally only lose two pieces of information: the flight identification (mode A & mode S data) and altitude (mode C data). Primary radar will still paint a target (albeit unidentified) as long as the aircraft is within radar range.
There's no fixed number – each individual is unique, as is the ejection that they endure. After ejection, a pilot will be given a full medical evaluation and it is down to that medical professional to advise whether it is recommended that the pilot continues to fly or not.