Does renewing the marriage contract with one's wife for reassurance mean nullifying the previous contract and divorcing her, and is it necessary to wait for the ‘iddah to end before the new contract, or is it permissible to contract immediately?

The actual integrity of the witnesses is not a condition for marriage; rather, their apparent uprightness is sufficient. If the witnesses' فسق (immorality/disobedience to Allah) becomes apparent after the contract, the contract is not invalidated by that. Therefore, marriage is valid with the testimony of those whose outward state is sound, because verifying the true integrity is difficult for the general populace. So, if it becomes clear after the contract that the witness was فاسق, it does not affect the contract.

Therefore, your marriage contract is valid, and you are not required to renew it. If you wish to renew it as a precaution, you are not obligated to observe استبراء (waiting period to ensure absence of pregnancy) or عدة (waiting period after divorce or death of husband), and menstruation has no effect on the contract.