You suggest everything and anything as translations for "wszystkiego". So can this sentence mean both
"He does not remember anything" (he remembers nothing)
"He does not remember everything" (he remembers all except a few things)
I would say that this sentence means "He does not remember everything"
"On nic nie pamięta." would mean "He does not remember anything" (he remembers nothing)
That makes sense because duolingo give both meanings. Curious that it can mean different things.
It shouldn't. "Nie pamięta wszystkiego" means "He doesn't remember everything" (but maybe he remembers 90%, who am I to say)
As mihxal said, if "He doesn't remember anything", that actually means that he kno... remembers nothing, zero, null.
"Anything" can be translated as 'wszystko', but in limited contexts, like "take anything you need" - "weź wszystko czego potrzebujesz".