Dusseldorf Weeze Airport to Stuttgart by night

How can we reach Stuttgart during the night and at a reasonable price?

You can't (unless you rent a car). That airport is 70km away from Düsseldorf in the middle of nowhere, and calling it "Düsseldorf Weeze airport" is basically false advertising.

There is a free shuttle service to the nearby Weeze and Kevelaer train stations, and a bus to Düsseldorf main station for 14 EUR, but only a single train to Stuttgart (at 2:47, with easy to miss connections) until next morning. See http://bahn.de for details.

My suggestion would be to spend a night in the airport hostel and make your way towards Stuttgart at your leisure the next day. However, a budget of 30 EUR per person to get to Stuttgart is only possible if you use the Quer-Durchs-Land-Ticket which is restricted to regional trains and requires chaging trains a lot (5 or 6 times between Weeze and Stuttgart).