Is it permissible to divide the making up of missed prayers (Qada') over different times of the day, so that some are made up, then one stops and resumes making them up later, while performing the present prayer at its appointed time? Or is this considered delaying the prayer? And is it obligatory to make up all missed prayers of a single day at once, in their proper order from Fajr to Isha?

If you have made up as many missed prayers as you can during the day and night, then there is no blame upon you if you stop at the 'Asr prayer or any other prayer to complete them the next day. It is not obligatory when making up prayers each day to start with Fajr and end with 'Isha. Rather, you make them up according to your ability, because these are missed prayers that are made up in order.