Was the marriage contract concluded correctly, Islamically, despite the absence of witnesses other than the guardian, the husband, and the officiant, bearing in mind that the husband is Sunni and the wife is Shiite and they have two children?

The marriage is invalid according to the majority of scholars due to the absence of witnesses, which is an essential condition for the validity of the marriage. As for the children, they are attributed to the father and inherit from him and he inherits from them, because they were born from intercourse due to a misconception (shubhah). The contract must be re-executed correctly with all conditions fulfilled. The absence of a written contract does not invalidate it, although it is better to write it down to preserve rights.