These specific fields are hardcoded in the layout so using a form plugin to move them around won't work. To make any changes you'd have to create a template override. Copy
And customize it to your liking.
In URL assignments section:
Set the option to Include
In URL matches enter /news/?$
Enable Use Regular Expressions
*The /? part is used for the case your /news page is also loading via /news/ and you need to show the module in both cases.
module_name.php file can't be overridden.
Check to see if the module has options to disable the loading of its scripts - Many modules come with such settings.
If it hasn't, then you could propose this to the developers of the modules to include it in the next release.
You can always clone the module and make one of yours that will behave the way you want.
The display of modules on a Joomla website is generally controlled via the "Menu Assignment" tab when editing the module in Module Manager.
Clicking on a tag link may take you to a different page so that modules displayed on the original page are no longer displayed on the new page.
Module assignment options are:
On all pages
Only on the pages ...