Welcome to the Q&A Forum

No, Google won't see it - you're meant to make your site accessible to those with JS turned off. Check using GWT 'fetch as Google'.

You could, however, do this with PHP.

What you have looks fine to me, I don't think there will be an issue with repeating 2-55 bags in the product title as it would be useful for differentiating that bag from another bag of the same name that isn't a 2-55, eg vintage-chanel-caviar-skin-bag.

You could definitely remove the /shop/ or /product-category/ if your site is a shop and nothing else. If you have a blog, and want a 'bags' category then keeping /product-category/ might be preferable to avoid confusion and potential duplication.

This is done automatically by Wordpress.

That image is the Menu admin where you set up navigation menus. It looks like you've set up a custom Home menu item/link, not a page.

Just create a new page called Home if it isn't under All Pages and publish it. Then in Settings > Reading set it to be the static front page.