2 days Netherlands 2 France 2 Switzerland or Austria or Italy
@LesleyA if your priority is Netherlands, France, Switzerland, I would aim for that as three countries in 10 days is plenty if you actually want to see something and not be too exhausted to enjoy your trip. After all, you want to be able to enjoy a coke or a glass of wine in a pavement cafe and watch the world go by, as well as hitting the big sights. Can you fly into one city and out of another?
You mention cheap, but how cheap? Europe is expensive and you'll be coming in July, which is peak season. We travel in Europe all the time and we budget around £100 per day for our family of four for food, but if you buy food at supermarkets or eat at McDonalds you could manage on much less. A French loaf, 2 large bottles of water and a packet of ham will cost less than €10 and keep you alive.
Your main cost will be accommodation, which can be very high in cities in July. I would check out hostels (which often have family rooms), and budget hotel chains. The latter are often in out of the way places though, on ring roads or by the airport, which if you don't have much time for sightseeing will just waste valuable time. Sleeping on the train is a good idea if you can get a private cabin with couchettes, but if you're trying to sleep sitting in an ordinary train seat I think that could really impact on your enjoyment of the trip.
Entrance to attractions is variable as some museums are free (check websites to find entrance charges), but things like boat and bus tours (hop-on, hop-off, which are great if you want to see a lot in a day), can be expensive. One day tickets for the Big Bus Tour of Paris, for instance, are €37.80 for anyone over 12. The same thing in London is £36.90 if over 12. Using the metro/tube and a guidebook would be cheaper, but you'll see less and it will be more stressful.
Yes, hostels are an option, backpacking is an option, many European teens do Eurail after they leave school and there are budget ways of doing things.