I'm sorry there has been so much back and forth with no resolution yet. I've looked at the logs you provided, but I don't see anything that indicates what the problem is.
I have taken the 2.10.0 community release of GeoServer (exact same version, build, git commit) that you are using, removed the PostgreSQL 9.1211 jarfile, and unpacked the plugin for GeoGig (with version 1.1.1 of GeoGig) and re-packaged the WAR. You can download it from the Boundless S3 bucket here:
https://s3.amazonaws.com/boundlessps-public/geoserver/gs-geogig/geoserver-2.10.0-war.zip. It is structured the exact same as the Community release of GeoServer 2.10.0:
Archive: geoserver-2.10.0-war.zip
Zip file size: 79279798 bytes, number of entries: 5
-rw-r--r-- 3.0 unx 79428780 bx defN 17-Jul-14 08:55 geoserver.war
-rw-r--r-- 3.0 unx 18008 tx defN 13-Sep-18 20:42 GPL.txt
-rw-r--r-- 3.0 unx 26610 tx defN 16-Aug-13 01:24 LICENSE.txt
drwxr-xr-x 3.0 unx 0 bx stor 16-Oct-28 09:52 target/
-rw-r--r-- 3.0 unx 333 tx defN 16-Oct-28 09:42 target/VERSION.txt
5 files, 79473731 bytes uncompressed, 79279024 bytes compressed: 0.2%
If you could try removing your current geoserver deployment (copy it some place safe if you have data in it you can't easily reproduce) and replace it with the one I linked and let us know if you still have issues, it would be greatly appreciated. You may have a "geoserver.war" file and a "geoserver" folder in your Tomcat "webapps" folder. Remove/backup both if you have them before deploying the WAR above.
If it resolves the issues, maybe try to do a diff between the jarfiles in your current deployment and the ones in the WAR I've provided to see if anything is missing or different, that may indicate where the problem is. If it does not resolve your issue, if you could try it one another system (different OS) to see if it works there.