I no longer have Painter IX installed however, in Painter X if you go to HelpAbout Corel Painter X, and also in subsequent versions, you can usually find your serial number listed in one of the areas. In Painter X, it's located in the area with the Application Copyright info. I'm guessing that your serial number is also available from somewhere in that "About Corel Painter IX" area...at least I hope so.
Regarding your question about the "akami" file, I'm afraid I haven't a clue. :(
Btw, if you've previously used the full version of Painter, you will probably not be too happy with Painter Essentials...it's an intro type of program.
Windows 7 Professional 64-bit/Intel Core i5-3570K @ 3.40 ghz/32.0GB DDR3 RAM/NVIDIA Quadro 600DAA Instructor: Introduction to Painter