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 : 

• Create the folder /wp-content/mu-plugins
• Make sure the /wp-content/mu-plugins is writable by your webserver user
• Go again to your wordpress administration page and check if you still have the error.

You still have the issue

• Unzip the content of wp-table-manager.zip file on your local machine
• Create the folder /wp-content/mu-plugins on your website if it doens't exist
• Copy the content of the /wp-table-manager/framework/ folder from the previouly unzip files to the /wp-content/mu-plugins/ folder in your website server
• Go again to your wordpress administration page, you should not have the error anymore

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 checkbox 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, 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.

Yes, the content is saved automatically in AJAX, so there is no need for any button to save. You'll get a green message that gives feedback on saving.

You have a parameter to disable that and add a "Save" button.