WP Table Manager uses it's own framework which is automatically installed in the /wp-content/mu-plugins
Sometimes this directory has wrong permissions and WP Table Manager can't install its framework.
To fix this issue :
You still have the issue
Yes, it can be or not, as an option in each table. The responsive mode is really advanced as you can define a priority for hiding columns on mobile sizes. When columns are hidden, a mobile menu with check box will be displayed to force the display/hide of columns.
The column size is fixed during the table edition and if you need to respect your content formatting, the responsive can be disabled on each table. If the size of all the columns is too large for the container you'll have an overflow and you'll be able to easily scroll on tactile devices.
Yes, the content is save automatically in AJAX, so there is no need any button to save. You'll get a green message that give feedback on saving.
You have a parameter to disable that and add a "Save" button.
Yes, you can insert table in editor, in any custom post type or widget. You can also edit the full table from your editor.
Yes of course! this component is built for that.
Themes are here to help you with the design but you're completely free to design your own.