Should the fornicator be flogged despite their claim of a Mut'ah marriage, and are their children considered children of fornication?

Mut'ah marriage is invalid. The prescribed punishment (hadd) is not applied to someone who contracts it if they believe it to be permissible due to the ambiguity of scholarly disagreement. Any children born from such a union are legitimate. However, if someone contracts it knowing it is forbidden, then the hadd punishment applies, and the children are considered children of fornication.