Q: What are the halachos of a partnership with gentiles regarding mitzvos such as ribbis, bechor, challah, mezuzah, etc.?
A: Partnership with a gentile does not allow collecting ribbis from another Jew or paying ribbis without a heter iska, unless the gentile partner takes personal responsibility for the entire loan or in a corporation owned mostly by gentiles (see discussion in The Laws of Ribbis 16:15-23).
Animals belonging jointly to a Jew and gentile are exempt from the sanctity of bechorah (firstborn). Nowadays it is recommended that a person with kosher animals make a partial partnership with a gentile to prevent the firstborn from becoming sacred (Y.D. 320:3).
Regarding dough jointly owned by a Jew and gentile: If the Jew’s share has the requisite amount, he is obligated to separate challah. (Y.D. 330:3).
The Rema (Y.D. 286:1) rules that a house owned (or rented) jointly by a Jew and a gentile is exempt from mezuzah (unless the Jew has a private room). Others disagree, but even one who follows the stringent opinion should not affix a mezuzah with a brachah (Aruch Hashulchan, Y.D. 286:2)