Is the Hadith: "Whoever preserves four rak'ahs before Dhuhr and four after it, Allah will forbid him from the Hellfire" authentic, and does making up missed pre-Dhuhr Sunnah prayers after Dhuhr or Asr include the virtue of the Hadith, or must they be observed before Dhuhr?

"Qada" (making up) according to jurists is performing an obligation after its prescribed time has expired. It serves as a rectification for a shortcoming or a recovery for what was missed. There is no sin in abandoning supererogatory (Nawafil) and sunnah prayers, but it is permissible to make them up after their time has passed. The Prophet, peace be upon him, made up sunnah prayers after their time had expired. And for whoever makes up the regular sunnah prayers mentioned in the hadith, it is hoped that they will be included in the promised favor, because "qada" is a recovery, and it is hoped for the one who makes up what is hoped for the one who performs on time. The mentioned hadith is authentic.