Is it understood from the hadith: "A prayer following a prayer, with no idle talk between them, is recorded in 'Illiyyin," that the ذكر (remembrance/supplications) of the prayer are to be said after the obligatory prayer and the regular Sunnah prayers?

The "laghw" (frivolous talk) mentioned in the Prophet's (peace be upon him) saying: "A prayer after a prayer, with no laghw between them," refers to vain speech, not speech in general. This hadith (definition) cannot be used as an argument by those who connect the supererogatory prayer directly to the obligatory prayer without reciting the prescribed remembrances. The Prophet's (peace be upon him) guidance was to separate the obligatory prayer from what follows it with specified remembrances and supplications. He forbade connecting one prayer directly to another and guided towards performing supererogatory prayers in homes.