If a marriage between a Jew and Gentile is not recognised, does that mean 1. their offspring are not recognised? therefore 2. the Jewish parent is not obligated to pay child support nor have any say in the decision making of the said children?

The offspring always goes after the mother, so only if the mother is a Jew then he is a Jew. According to the civil law the father is still obligated to support the child even if the child is not considered his Halachic offspring. Rabbi Yitzchak Grinblat