Is the witnessing in the marriage contract a testimony to hearing the woman's consent to marry a specific man, or to her deputizing her father or guardian, or is it merely a testimony to the performance of the contract without requiring to hear the woman under any circumstances?

The jurists have stipulated that witnessing the marriage (nikah (definition)) is a condition for its validity. As for witnessing the woman's consent, it is recommended but not a condition. It is recommended when her consent is considered, such as when she says "I consent to it" or "I permit it," this is as a precautionary measure to prevent her from denying it. The witnessing intended is for the offer and acceptance between the contracting parties.