Is it permissible to pray the Sunnah (recommended) prayers that precede the obligatory prayers at times distant from the performance of the Fard (obligatory) prayer, such as praying the Sunnah of Dhuhr (noon prayer) then performing the Fard later after being occupied with another task, knowing that the Sunnah is prayed after the time for Dhuhr has begun?

There is no objection to separating the performance of the pre- or post- [obligatory prayer] supererogatory sunnah prayer from the performance of the obligatory prayer; rather, this is the Sunnah. The Prophet, peace and blessings be upon him, "forbade connecting one prayer with another until one speaks or leaves." Therefore, there is no objection for the questioner to pray the sunnah prayer for Dhuhr and then engage in something else before praying the obligatory Dhuhr prayer.