“He slack off his way through the course and still managed to pass.” “He slack off through the course…” which one is correct?

Neither. Slack needs to be in past tense as slacked.  I also might phrase it as "he slacked off throughout the course..."

Neither. It should be "He slacked off all through the course". Make sure you use past tense: slacked, not slack. You could also say (mildly vulgar) "He half-assed the course and still managed to pass".

Neither is correct. I’d say “He slacked off through the course and still managed to pass,” although “…but still managed to pass,” sounds even better to me.

He slack**ed** off. He slack**ed** (no off (it doesn’t feel right having it here to me)) his way through the course and still managed to pass. He slack**ed** off through the course…

would say he slacked off throughout ..