How to find the history map data set, e.g. the data of China in 2006
Very little will have been mapped in China in 2006, so you wlll be going back in terms of level of detail not in terms of things that have since been demolished.
I don’t believe there is any API that will give you the historic contents of a bounding box. The closest you might get is old changesets, but doing that on even a small country would, I think,be an abuse of the APIs.
The big problem is deleted objects. I think you can only get information on those if you know their ID.
What are you trying to achieve by doing this?
You can download the Full History planet or Geofabrik have history files (.osh) for some regions. However I believe these are ODBL datasets so only go back to September 2012.
To process OSH files you need to compile Osmium.
The alternative is to find an old planet file: a 2006 one will be relatively small & cut out China using osmosis or osmconvert. They seem to vary in size between 100 and 300 Mbytes. I dont know what quality is like or whether the format is readable with modern tools. Files are here: http://planet.openstreetmap.org/cc-by-sa/.
With https://mvexel.github.io/thenandnow/#7/40.036/120.070 you can have an idea how much data there was in 2007. After a quick look, only Bejing had some data in 2007.