How many times have/can you find it? and in how many different locations?

I've found it once per game in random locations.

The problem is that the quest respawns. Just don't ask the Jarl about work, or whatever.

I found it 2 different times in 2 different locations, then picked it from the jarl and kept it.

For a while I was living at the college and finding a new helm every few days... Always in a new spot.

The only thing I know is evidently a number of bugs arise if it's located in the Yngol Barrow and Yngvild, so you should save before you ask him for the quest and reload if you get those two.