Should the current Asr prayer be offered before a missed Dhuhr prayer? Is it permissible to make up for missed regular Sunnah prayers between Maghrib and Isha? Will someone who missed the opening Takbir and the first row due to being overcome by sleep be rewarded as much as someone who caught them?

If sleep overwhelms you, preventing you from catching the takbirat al-ihram (definition) and the first row in congregation, and Allah knows the sincerity of your desire, then it is hoped that you will attain their reward. The order between the two shared-time prayers (Dhuhr and Asr, and Maghrib and Isha) is recommended. Performing Asr before Dhuhr does not invalidate the prayer; rather, some scholars believe that attaining the congregation supersedes the obligation of maintaining the order. As for making up missed Sunnah (definition) rawatib (supererogatory prayers), it is not obligatory according to many scholars. The correct view among the Shafi'is is that it is recommended to make them up at any time. Thus, it is permissible to make up the Dhuhr Sunnah between Maghrib and Isha.