“gotten” is chiefly American English past participle of “get”. Standard British English uses “got” as the past partictiple. Both use “got” as the past tense.
Your example is not correct English (in either style), and I am not very clear what you are intending to say. You may mean something like “…so that he can know how the results of the program turned out.”
Not exactly. In that case you could possibly say “…so that he can know/discover/learn/find out where all the results produced the program come from.” This would usually be asking about the program’s internal workings and/or its data sources.
“can know” is not a massively common combination, and it may be that one of the other alternatives is a better fit.