Yes you can create a woocommerce folder inside your theme, copy the templates over to the theme and modify them.
If you do this, make sure to use a child theme though, that way if the parent theme you are using gets updated you won’t lose all your WooCommerce edits, since updating your parent theme will delete all the files and replace them. Having your edits in a child theme will keep them safe for any future theme updates.
That said, WooCommerce has TONS of hooks and filters available so generally you do not have to modify any template parts you can do most edits using the filters/hooks and custom CSS.
When editing the templates you need to make sure to review them every time WooCommerce releases an update incase there are important security or bug fixes. So editing the templates manually can add a significant amount of extra work in the future keeping things up to date and secure. That’s why using filters, hooks & CSS is ideal.
In terms of doing things like “adding sidebars” that’s something that should be a part of your theme by default and shouldn’t require modifying any files. All you need is to have a woocommerce.php file in your theme or child theme as this is used for all WooCommerce pages so you can define the sidebar and core HTML there. This is explained in the docs here – https://docs.woocommerce.com/document/third-party-custom-theme-compatibility/
I wrote a little post with some great snippets for changing some core functions that may be useful to you – http://www.wpexplorer.com/woocommerce-compatible-theme/