Is the aforementioned marriage contract valid, and what must be done if it is not, especially since the offer and acceptance did not occur before the Notary Public, and one of the two witnesses was not present at the عقد (contract)?

A legitimate marriage has conditions and pillars without which it is invalid, including the guardian (wali), witnesses, offer (ijab), and acceptance (qabul). If the marriage takes place without an offer and acceptance between the husband (or his proxy) and the woman's guardian (or his proxy), or if two witnesses or one of them are not present at the contracting session, then the contract is invalid according to the majority of scholars. It is not a condition for the validity of the contract that the offer and acceptance occur in court or before notaries. If the marriage does not fulfill the conditions and pillars, it must be renewed with an offer and acceptance between the husband and the guardian in the presence of two witnesses.