Unzip the installer and install things 1 by 1 to see if there are any errors. Everything is inside the "extensions" folder of the zip. There is something wrong with your CB install or Joomla install. I've retested a full package (all our products) 3 times now and it works perfectly fine. You most likely have a permissions problem preventing files from moving properly. If you're installing through Joomla extension manager try installing through CB plugin manager or vise verse as needed (our installer works fine in both).

- errors trying to access CB in admin are :

I don't see how those errors are possible for a clean install. That configuration file does not exist and is only included if it exists to migrate CB 1.x configuration. Your install should not have that. Especially if it's a clean install as you stated.

and we got the following log from CB installation:

The plugin errors are saying the access column does not exist. This means your CB database is not CB 2.x, but still CB 1.x. Again, not possible if this is a clean install.

this is a clean install as a clone of our live site. CB has been migrated to CB 2.0 a few months ago and is working great on the live server.

A clone is not a clean install. Deploy an empty Joomla to your localhost then install the package you downloaded. If it works fine it's a conflict of some sort going on on your clone/live site specifically.

Unzipping CB packages to install only package 001_pkg_communitybuilder_2.0.14+build.2016.06.16.19.39.02.18f2ce0c8 works fine and installs correctly.

CB was installing fine in your package. Try installing one of the plugins that's throwing an error.

I will have a dev clone of the live server available in a couple of hours if you want to have a look

That'd be a huge help, thank you. Please PM login credentials once cloned and will debug the package installer. Also note I am already working on improvements for the installer to better output errors.

Attempted to install a package of only CB Auto Actions. It then proceeded to try and install a very old package builder package. This is most absolutely Joomla tmp folder issues. Using FTP completely empty the /tmp folder in Joomla root (do not delete the folder). This has indeed crashed the test site due to the old code being installed.

I'll need to figure out a way for the package installer to protect against this. For now just install the packages through CB > Plugin Management and you shouldn't have these Joomla tmp folder issues.