Is agreement, a proposal of marriage, the guardian's consent, and public declaration sufficient to complete the marriage contract, and is the girl considered his wife, making it permissible for them to talk and go out without a mahram, or is documentation in court required?

The wording mentioned in the question does not constitute a valid marriage contract; rather, it is a proposal. A suitor is not permitted to be in private seclusion with his fiancée. A marriage contract is concluded through offer (ijab) and acceptance (qabul) using phrases that signify it, such as "I marry you (zawwajtuka)" and "I accept the marriage (qabiltu az-zawaj)," and the presence of two witnesses is required. The validity of a marriage does not depend on its registration in courts, but it has become necessary to safeguard rights. Registration is a matter mandated by law to protect the contract from denial and repudiation after its conclusion.