Eurostar is expensive at 30€ per seat; Thalys is comparably expensive.
Amsterdam to Berlin on 17/6 at 5 am is no problem on the DB website (4€, you can choose yours seats from a seat plan) or ÖBB website (3€, no seat plan). There's no direct night train.
Prague to Salzburg on 19/6 at 6:04 can be done via DB if you enter Freilassing as destination. It will then let you make reservations Prague - Linz and Linz - Salzburg for 4€ per person in total. (*)
Salzburg to Venice overnight on 19/6 can be done via ÖBB using these steps. The night train only leaves from Salzburg at 01:40, so you might consider to go to Attnang-Puchheim, Wels or Linz to enter it earlier. Or take the 8:12 train on 20/6, with a change in Villach. Again, you can book the trains on 20/6 for 4€ via DB if you enter Freilassing as your starting point (via ÖBB it's 2 x 3€ = 6€). (*)
Venice to Rome on 21/6 is best booked via ÖBB using these steps but the Italian summer timetable hasn't arrived there yet. Shouldn't take long anymore. You can already check the timetable on www.trenitalia.com (which is the first source for Italian timetables): the first direct train leaves at 5:26, the next one at 6:26. Reservation via ÖBB should be 10€ per seat (Interrail will charge 12€). There's no big rush though since these pass holder seats are not quota-controlled.
(*) DB will only let you book reservations if they can price the journey. And they can mostly only price the journey if it's from or to Germany.