We went to the Dutch immigration desk/office in the terminal (just to the right of the control area you pass through to clear immigration). They will issue your child a transit visa as long as you will be departing again within 24 hours. The whole process took about 10 minutes.
A couple of hotel suggestions - there is a Courtyard Marriott offsite but with a free shuttle that you catch just outside the terminal. Shuttle runs on a regular schedule. There's also a Sheraton right at the airport - no shuttle, a short walk from the terminal (there's a pedestrian bridge). You can book either on line. there are lots of other hotels in the area, check out
http://www.schiphol.nl/ and search on "hotels"
The airline (NW/KLM) checked our bags all the way through in Moscow, so we didnt need to claim luggage in Amsterdam (just kept an overnight bag with us). They also gave us our boarding passes in Moscow for our flight back to the US, so we were able to skip the check in line in Amsterdam and go straight to security when we were leaving