Checking for plugin updates (Through the GUI) should only take ~30 seconds
Jun 9 00:00:01 Tower Plugin Auto Update: Checking for available plugin updates
Jun 9 00:00:31 Tower Plugin Auto Update: Community Applications Plugin Auto Update finished
Since you've got CA Auto Update installed and running, I wouldn't particularly worry about ever checking for plugin updates, as the Plugin page will show automatically whatever CA found, regardless if you have CA automatically install the updates or not.
Same thing goes for docker updates. If CA is configured to check for docker updates (regardless if you're installing the updates or not), then the docker tab will always be up to date based on when it last ran.
Not sure though why checking for updates on the docker tab actually times out.