Compatibility patch only works for COH v2.602, so if you tried it in the last version of COH it will cause problems for you.
The log that would help us would be warnings.log that is in the previous folder where you find all the ones you posted. So far, the bug you are getting was removed in the last version of EF as far as I remember, so probably not all the files were updated if you installed EF v2.30 in v2.10.
I'd suggest to uninstall EF, then go to the COH folder and delete all the remaining files if you find them (EF launcher files and folders, EF mod files and folders, FactionFix.dll). Then install 2.30 from scratch.
If your COH is version 2.602 then use the compatibility patch, otherwise just enjoy EF.
If the "unable to locate coh directory" error persists, you would need to create a shorcut to run it. If you still have problems please let us know.