If there is no EA background service then EA APP is not installed correctly. Reinstall EA APP.
If the program did not create anything, do not create these folders yourself.
Don't create anything manually, I know cases from the time of Punkbuster where people received global bans for manually modifying anti-cheat files.
Unlock the built-in administrator account and try running BF2042 there. The built-in admin account does not have any NTFS permission restrictions on the C drive, so this may help.
I had a case where BFV did not save my single player progress due to the lack of permissions to modify its files on the C drive on a regular admin account (made by user). The problem disappeared with the built-in admin account.
From then on, I only use the built-in Windows administrator account. I am deactivating the admin account I created (standard admin).
Maybe for the same reason EAAC does not install correctly. "Run as administrator" mode is not the same as working with the built-in administrator account in Windows.
Below are the screenshots you asked for
GL
