On average, how often (in flight hours) will a pilot face an emergency situation?
The answer is going to vary wildly according aircraft type, pilot rating and proficiency, number and type of engines, aircraft size, and part of the world. 

In this 2016 study by the FAA for GA pilots, the median period of time for non instrument rated pilots was over 3 times as often as instrument rated. 250hrs vs 823hrs. The report describes some of the hardships involved in arriving at the less than precise determination.

I have asked this question to many airline pilots and about half of the 4000-6000hr pilots have a story to tell. Just guessing but I would say modern airliners are perhaps every 10,000 hours.

I must be doing something wrong, because I averaged about every 500-600hrs but then I was mostly flying very old run down Beech-18's in Alaska.