Is it permissible to work as a programmer for a company that sells permissible household items, but also sells prohibited items like artifacts and paintings that may contain crosses, knowing that the programmer's work will not directly deal with the products?

It is permissible to work in programming that is used for permissible things, such as sales and purchase programs, due to the general permissibility, as long as it does not involve aiding in what is forbidden, based on the verse: (And cooperate in righteousness and piety, but do not cooperate in sin and aggression). Therefore, it is not permissible to create a program for an interest-based bank or a liquor store. As for stores whose main sales are permissible, it is permissible to create a program for them, even if they sell a small amount of forbidden items, which is a widespread affliction, provided that the program does not explicitly mention any of the forbidden items.

It is not permissible to sell statues, crosses, images of living beings, or images containing crosses, because it is not permissible to create any of these things. Rather, it is obligatory to efface images and statues and to destroy crosses, and indeed, when Allah forbids something, He also forbids its price.