Is it sufficient, when intending to pray the regular Sunnah prayers such as the pre- and post-Dhuhr prayers, to generally intend to pray Dhuhr prayer, or must one specify that it is the pre- or post-Dhuhr prayer?

There must be an intention for the regular (Sunnah) supererogatory prayers. A person should intend to pray the regular supererogatory prayer before or after Dhuhr, and the intention for the obligatory Dhuhr prayer is not sufficient for the regular supererogatory prayer. Ibn Qudamah said: "Specific supererogatory prayers, such as the eclipse prayer, the prayer for rain, Taraweeh, Witr, and the regular supererogatory prayers, also require specification (of intention)."