The chlorine is used to kill the bacteria.
It will not show up in a water test because it is not a health/safety issue.
Hydrogen sulfide is the rotten egg smell produced by the bacteria.
The carbon filter is helping remove some of the smell/taste issues.
Because the bacteria are naturally occurring... the chlorine shock wears off and they re-enter the well water.
Each time you ''shock'' the well, it kills enough bacteria that the carbon filter can handle the small amount. But once the bacteria begin to multiply again, they overcome the carbon filter at some point.
https://extension.psu.edu/hydrogen-s...in-water-wells