Do you want to come home with me and listen to music is the most idiomatic in English and it was accepted. It would have been funnier to me if the offer was to see his etchings though.
Perhaps only to Americans of a certain age. :-)
Home to my place is redundant and,since I'm not a native English speaker I'm not sure if it is even correct. Come to my house would be much better I think! So I translated verbatim and it was declared wroong!
The multiple choice questions are created randomly so others won't get the same alternatives, and maybe they won't even get this as a multiple choice question. Generally, du is the singular and ni is the plural.