Ten days lets you string together Beijing, Xi’an and Shanghai if you base cleverly: three or four nights per city, one overnight on the move by high-speed rail, and licensed hotels near the metro in each. This plan treats sleeping well as the backbone, not an afterthought.
A foreign passport holder traveling China with kids should sleep in four family-friendly bases: three nights in Beijing (Dongcheng), two in Xi’an (inside the wall), three in Shanghai (Huangpu), and two in Guilin/Yangshuo. Each base has licensed hotels with proper family rooms and registers foreigners cleanly — the key is booking connecting or extra-bed rooms in advance and confirming the foreigner policy in writing.