I just adopted a child in Skyrim, and they're staying in the second bedroom in Breezehome (your house in Whiterun). They left me a few items in the chest, but I'm wondering if it's possible for them to steal back (or randomly disappear) items that are left in that chest. In other words, is the chest in the second bedroom "safe" if I store items, or should I always store them upstairs, in the master bedroom?
It heavily depends on the house, and what specific containers. Breezehome in specific is safe. You can check this link: http://www.uesp.net/wiki/Skyrim:Breezehome for the status of different houses, in regards to chest and container safety. This specific house does not have re-spawn containers, so it is safe.
All containers in your house are safe, even childrens containers since they are part of your house. You don't need to worry about any container in your house. They're all safe since the house is yours.