There are two problems with Toph's inability to break her and Katara out of the wooden cell. First, she should be able to bend the wooden bars, as the beetle merchants from Book Two's "Appa's Lost Days", also Earthbenders, were able to bend the wooden restraints on the their sand sled. Secondly, both girls are wearing clothing accessories apparently made of gold that Toph should be able to bend into something to cut their way out with.
After the Toph and Katara argument about Katara's "motherly" attitude, all four are clearly shown walking along a street in the town. But in the next shot, when Toph points out where they can gamble for money, Katara is gone, and remains absent for the rest of the scene.