Adding late fees automatically

It is "automatic" once you open the "Assess Late Fee" function, from the Home page, the Customer menu, or the button on the right side of Create Statements.

You mark who to assess. The Program doesn't know whose payment just came in the mail, so there is no Automatic = Everybody, because that is not a safe assumption. It will show Everybody checkmarked, and you need to review this screen. it also Flags * for those customers with open AR balance and Available Credits, to remind you to apply their credits, in case they are not really Late so much as "oops, I forgot they prepaid."

"Is it possible to add a 10% late fee for any unpaid balance a customer may have after the grace period and a 15% late fee for each following month automatically?"

The Late Fee would be One Rate. It cannot Vary and still be Automatically used. You can set a Minimum Charge, along with a Rate, though.

Please see my attachments.

Not automatically.  You do this using finance charges.  Somehow I doubt 15% per month is legal.