Does the rule not to eat Chametz after Pesach that was owned by a Jew on Pesach extend to things, like soda, which are only possibly Chametz? How about food that was owned by a company, that is part owned by Jews and part owned by gentiles?

Only real Chametz is forbidden after Pesach if it was not sold, and not products that don’t have supervision but have no apparent real Chametz. In regards to a partnership with a non Jew, the Mishna Brurah (448;2) assumes that the portion of the Jew is forbidden and if the partnership has not been dissolved, all the Chametz is forbidden. However there is another opinion (Sha’agat Arye) and the ultimate decision will also depend on what type of partnership (private, incorporated, shareholders, etc.) A decision would have to be made on individual cases.