I'd like to point out that you are running GameEx Arcade 15.10 and I am using the latest version of 16.07. Some things may have changed between yours and mine. I am not saying that you have to upgrade, but it makes this more challenging. I would not ask you to upgrade except as a last resort or unless a reload is suggested.
The first thing I see in the log.txt is that each emulator other than the integrated Mame emulator has an error stating that the mapfile does not exist. You will find these values in the Advanced Emulator Setup sections for each of the emulators. In GameEx Arcade 16.07, there are no mapfiles included in the install by default; not sure on 15.10. I suggest you recheck the mapfile paths on your install to see if the paths and filenames are 100% correct.
I do not see anything wrong with your emulator configuration paths or command line unless the paths where your files actually reside have changed.
One thing I suggest is to change your command line in the Emulator Setup for each of your emulators to the following (you have the same flags enabled on the Integrated Mame, so I assume you would want to be consistent):
mame.exe [ROM] -rompath "[ROMPATH]" -nowindow -joy -skip_gameinfo
You also mentioned trying to add an Atari emulator, but there is no Atari emulator in your config. Where should I see it? Emulators 1 through 5 are there and they are all Mame driven emulators.