How to create specific role permissions for authenticated users in Drupal 7
You could have a look at User points module, you also need userpoints_nodelimit, part of User Points Contributed modules to limit the number of nodes a specific user is allowed to create. For the deletion of nodes you'll need the rules modules so you can give the user a point back.

You could write your own implementation, but depending on your skills this might be harder, in short you need to implement:


an access callback where you check the user and the number of available points
use hook_menu_alter to replace the access callback of 'node/add'
implement hook_node_delete and hook_node_isnert to change the counter