What will be the Posts Feature image dimension?
Recommended image size larger than 420px for perfect view on various devices.

Do I need to regenerate thumbnail images of existing posts?
Yes. But it is optional. You can use the Force Regenerate Thumbnails plugin or any other plugin of your choice.

How to use shortcode inside page templates? 
WordPress has a significant function, do_shortcode(), that will allow you to use shortcodes inside your theme files. For example, to output posts in a Theme file, you would do this:

<?php echo do_shortcode('[gs_wpposts theme="gs_wppost_grid_1"]'); ?>