Every document in Odoo, including a check, retains both metadata about the user who created it and the date it was created, as well as the date and user who last modified (edited) it.

There is nothing preventing users printing checks, but you could easily add security restrictions to prevent certain users from printing checks.  

Recording how many times a check has been printed (and who printed it) would require customization.

You can use Audit Trail module... which tracks any changes made to record by anyone... however this module is available for version 6... so if interested, you can modify that module and make it work in Odoo....