C.,
the best bet to get a hotel you are used to like in the states is guest housing--but they will only book you about 7 days out. Only official travel (PCS, TDY) is bookable beforehand with date. you can get a hotel and then on short notice try one of the 3 guest houses Stuttgart area (though Robinson is a bit harder when it comes to public transportation use).
Public transportation is in Stuttgart plentiful.
On the stuttgart web site I think there are hotels listed that give you a "deal" when you book for the Christmas Market and you get perks like a public transportation ticket included for 3 days eg. See what you can find out through that site.
When you go into the Rathaus in Stuttgart (main market place) you can use the elevator to get to the top floor and see the market from up top. Is especially nice at night when the lights are on.
In the Rathaus there is also free childcare available(they like donations though) from about 2-6pm weekdays (age I don't remember though. They will take 3 years but I'm not sure of the 2 year old.)
Have fun travelling,
E.