That is incorrect. Reservation is required for Nightjet trains and also a very good idea because they are often fully booked well in advance.

It's best to book via ÖBB (no booking fees). Do NOT use the "Seat only” option but add the "Interrail / Eurail” discount and then look for normal tickets. There, you should also have more choice in the available accommodation, e.g. also Double and T3 for the sleeper.

Thank you so much, I had no idea. 

Can you tell me if this one is in the same situation? 

The IC Budapest - Zagreb has optional reservations. You can find that in e.g. the DB planner, which is a much more reliable source. Reservations can be booked for this train via ÖBB (3€) now using the "Seat only” option.

OR do it on a yellow machine by MAV when In HU-cost is less as 300 HUF (=less as 1€)-do it only till last stop in HU-you sit on that seat then anyway. Again: NO, these trains are not fully booked weeks before.

I also have some doubts about that Budapest-Zagreb train. The thing is, what happens if there’s no more seats available at the moment of departure? Can we just seat on the floor? Or do we need to get the next train?

Most probably, just like in most other trains. Remember that for most trains a reservation is optional or not possible. But AFAIK it's very rare for this train to be overcrowded.

The hungarian timetable for this train says: “In domestic traffic seat Reservation compulsory, seat Reservation in international traffic can be made.” - So you may board without a seat reservation.