In case of delays you need to get in touch with the staff of the train and ask them to give you some prove or something to get on next trains without needing a new day. If you travel with the same company probably you don’t need such a proof, but better to ask. 

Keep in mind, that in case you can’t get the last train of the day, you are allowed to get from the train company a Hotel room or taxi (maximum 50-80 EUR, depends from company to company), sometimes you need to pay it yourself and get it back by the company, but I don’t recommend it. 

It is the departure time according to the time table that is relevant for the Interrail/Eurail app. It doesn't matter if the train gets delayed. If the train is scheduled to depart before midnight, then the travel day is the day before midnight even if the train actually departs after midnight.

You only need to remember to change the date in the app where you show the QRcode so that you find the QRcode for the previous day. 

This is difficult to be definitive and I would think it depends on whether you are planning to board a connecting train scheduled to depart before midnight. If you miss that because your first train is late then it is the responsibility of the first train operator to sort out an alternative train at no expense to you. simply get them to give you a voucher,then I would add my original train to the pass and use that with the voucher to support any inspection. 
I believe if there is no alternative train that night they need to supply accommodation and a no cost onward train in the morning, but others will correct me if that is not correct.

If your connection was always planned to depart after midnight then the same applies except you would still need the second travel day.

Note that Interrail app works on scheduled departure time so if your train is scheduled to leave before midnight but leaves after midnight it is still the original travel day (But if you have added the train manually make sure you add the scheduled departure time not the actual departure time).

Best is-a nobrainer- to not plan that tight. Aso note that on the very last day of validity of pass the pass does end at 23.59. So you are not allowed on THAT day to even use (in theory) a train that arr at f.e. 0.11. Though often in such countries were trains run till that late they do not mind at all.

The very last trip is for 99% to go home-and you likely know best how well that railway is in timekeeping-and also in giving compensation. Often there is a very wide cliff between theory and practice. F.e. in NL HTL is hardly ever paid for-you have to contact NS and they promise to arrange to bring you home-most often by taxi, together with others. Taxi to be ordered by NS.

In german forums you can read endless reports and more about people needing to go to court to get their claims even recognised-that why I started with: you will not want that.

Thanks to all of your for your very valuable answers to my question.

@mcadv: Thank you for your advise. Yes, I know. Yet, nowadays even when you’re planning with having loose time schedule in mind, things still may get vexing in Germany. For my question, I was interested in receiving a general regulation rule, so I may claim my rights to the train attendent for being allowed on board if things go weird.

For me, it’s particularly useful to know the rules if you miss your connection and need to take a subsequent train connection after 12 a.m.