Can Sharepoint 2010 and Sharepoint 2013 use the same database (SQL 2012) ?
Yes you can leverage the same database server across multiple installs.
The only thing you should take care of is to use unique "Database" names so it will not conflict with each other.
Append "SP2010" to all databases that are created for SharePoint 2010.
Also you should increase the server capacity to support both.