Is the Dhuhr prayer, which was combined with Asr, invalidated if one prays Asr again after the necessity for combining them is gone? And in this case, should Asr be intended as a make-up (qada') or a supererogatory (nafl) prayer?

If Dhuhr is combined with Asr due to a legitimate Shar'i excuse, and then the excuse ceases before the time for Asr begins, the combining is valid and there is no obligation to repeat Asr. This is because the prayer was performed correctly, sufficiently, and the obligation was fulfilled. Also, because one performed their duty while the excuse was present, it is not invalidated by its cessation. If the Asr prayer is repeated as a supererogatory act, the combining is not invalidated.