Is it permissible to delay the regular Sunnah prayer from the obligatory prayer, such as the Dhuhr prayer, so that it is performed at home sometime after completing the obligatory prayer, or is it better to perform it immediately after the obligatory prayer at home?

It is permissible to delay the post-obligatory Sunnah prayers until the end of the time for the obligatory prayer. Their time extends from the performance of the obligatory prayer until its time expires. Performing the regular Sunnah prayer at home is better than performing it in the mosque. Hastening to perform it immediately after the obligatory prayer is better than delaying it, for fear that some impediment might prevent its performance and thus its reward be missed.