... I don't know how to save a search query into SOBI2 database and display it like 'Latest Searches' for example. ...
As far as I know what you require is not possible with the standard SOBI2 or SOBI2 Search Module.
SOBI2 Searching is done using
Ajax technologies and the Web Server does not store or log search histories.
Note: The URL does not change when you are using the SOBI2 Search function.
It is possible to create Joomla! Menu Items with pre-defined SOBI2 searches (refer
http://www.sigsiu.net/special_links.html - sobi2Task=search), but I do not think this capability satisfies the requirement you describe.
Regards - Mark