Dan,
as the majority of your error log lines are pointing to an issue with the Moderator component,
my first idea would be to temporary disable it and verify whether this makes a difference.
Next, please be more precise about which buttons / actions exactly are causing that trouble.

Also Running 4.4
Exact same issue... error logs below:

[01-Oct-2017 02:46:20 UTC] PHP WARNING: 2017-10-01 02:46:20 (UTC): "exifreaddata(phpssbvAW): Illegal IFD size" in file /home/user/publichtml/classes/OssnFile.php (line 452)
[01-Oct-2017 02:54:45 UTC] PHP WARNING: 2017-10-01 02:54:45 (UTC): "Cannot modify header information - headers already sent by (output started at /home/user/publichtml/components/OssnGrou$
[01-Oct-2017 02:58:04 UTC] PHP Fatal error: Cannot access empty property in /home/user/publichtml/components/Moderator/actions/group/cover/delete.php on line 29
[01-Oct-2017 02:58:08 UTC] PHP Fatal error: Cannot access empty property in /home/user/publichtml/components/Moderator/actions/group/cover/delete.php on line 29
[01-Oct-2017 02:58:12 UTC] PHP Fatal error: Cannot access empty property in /home/user/publichtml/components/Moderator/actions/group/cover/delete.php on line 29
[01-Oct-2017 02:58:16 UTC] PHP WARNING: 2017-10-01 02:58:16 (UTC): "Invalid argument supplied for foreach()" in file /home/user/publichtml/components/Moderator/actions/group/cover/delete.$
[01-Oct-2017 02:58:18 UTC] PHP WARNING: 2017-10-01 02:58:18 (UTC): "Invalid argument supplied for foreach()" in file /home/user/publichtml/components/Moderator/actions/group/cover/delete.$
[01-Oct-2017 07:33:22 UTC] PHP WARNING: 2017-10-01 07:33:22 (UTC): "Invalid argument supplied for foreach()" in file /home/user/publichtml/components/Events/actions/relation.php (line 30)
[01-Oct-2017 21:03:05 UTC] PHP WARNING: 2017-10-01 21:03:05 (UTC): "exifreaddata(phpPz87w7): Illegal IFD size" in file /home/user/publichtml/classes/OssnFile.php (line 452)
[01-Oct-2017 21:10:02 UTC] PHP WARNING: 2017-10-01 21:10:02 (UTC): "exifreaddata(phptvYKcN): File not supported" in file /home/user/publichtml/classes/OssnFile.php (line 452)
[01-Oct-2017 21:10:07 UTC] PHP WARNING: 2017-10-01 21:10:07 (UTC): "Cannot modify header information - headers already sent by (output started at /home/user/public_html/components/Business$

"Running 4.4 OSSN".
I am not able to locate either of these logs.
I also tried in latest FireFox and IE browsers. Seems the JavaScript is not executing properly. Click on the buttons and nothing happens. Only the change cover button works.

Can you tell what version of OSSN you are using? try to look into error_log file and try to view browser console log.