"my family." This probably does not refer to a physical house. The rich man wants Lazarus to warn his family members, even if they do not live in the same house as his father.
## in order that he may warn them ##
This could be translated as "Tell Lazarus to warn them."
This could be translated as "so that they do not also come" (UDB) or "If they are not warned, they might come." Since it is implied that the way they would avoid going there is by repenting, this implied information could be made clear, as in "so that they might repent and not come." (See: [[rc://en/ta/man/translate/figs-explicit]])
This could be translated as "this place where we suffer torment" or "this place where we suffer terrible pain" or "this place where we are tortured." (See: [[rc://en/ta/man/translate/figs-explicit]])