Hello! I am looking into whether the 5-day pass is also valid if I start the journey on, for example, July 23, and arrive at the destination on June 24, but there are also transfers in between. In such a situation, does one day or two days already count?

Assuming you mean July 23 and July 24: It depends if the transfer is before or after midnight Central European Time. The use of travel days is determined by the moment you board the train (according to the timetable), so if you board a subsequent train after midnight you’ll need to use a travel day for that day as well.

When buying a train ticket from Milan to Nice, does the ticket include transfers?

There are no direct trains from Milan to Nice, so you will have to transfer. You have to buy passholder seat reservations for the direct trains between Milan and Ventimiglia, but with your pass you don't need reservations for the TER regional train from Ventimiglia to Nice. The ticket is your interrail pass, which does allow taking both trains on the same day (with the added need of a reservation on the Italian train).

There is also a question about the line between Venice and Rome, as well as Rome and Athens. Does this line exist? This option is not shown in the search.

There generally are plenty of trains between Venice and Rome. Depending on when you’re intending to travel they might not yet have been loaded in the timetable. Check on trenitalia.com selecting a date in the near future to get an indication of what's available.

Checking for 15 February for instance shows there are hourly Frecciarossas (passholder reservation €10), and some other options (such as an Intercity, €3 reservation fee):

Once the timetables for your travel date become available, you can book them through the Austrian railways. See the instructions here:

Between Rome and Athens it is a bit more difficult: if you look on a map you will see there is a sea in between. You can cross to Greece by ferry from Italy (not from Rome, so you will have to take a train to the other side of the Italian peninsula).

As such GR is currently a rail-island-all its former overborder lines have been closed, at least for last part till border. Best site for help in seeing what one can do is seat61,com

There are NO overnight options for Nice-Milano, if you saw this it will mean a long wait at some minor stop with probably a closed station. Not advisable. Ventimiglia as such is even worse. I think as for now it is even impossible to buy a ticket (normal full fare) from MIl to Nice all the way-you have to buy a new FR for the TER train in Venti. Once that was very common, but modern tech does not help always.

I am unsure why you feel somebody with a Global pass would need any ticket from Milan to Nice, but there is absolutely no problem booking that route as a through booking, but it obviously requires a change at Ventimiglia. You do of course require a reservation (3 euro) for the direct Intercity service Mil - VM.

OOI the cost of a single Mil - Nice is about 38 euro.

The complete journey on the IC/TER service is typically 5 hours so why would a night train be needed anyway?