For your FrontendPlugin you probably want to add your JavaScript and CSS with TypoScript. In extbase extensions this is normally done in Configuration/TypoScript/setup.txt:
page.includeCSS.myExtension = EXT:my_extension/Resources/Public/Css/MyCss.css
page.includeJS.myExtension = EXT:my_extension/Resources/Public/JavaScript/MyJs.js
As for your 2nd question. You specified the same default Controller/Action for both your frontend plugin and your backend module.
array(
'Weather' => 'list',
),
So both are using the same action wich is using its template.