[Storefront] Header size and Header widget areas

Hello All,

I’ve been trying to modify storefront theme header for a couple of weeks now, but haven’t found my way around what I exactly want.

I simply want four things:

1. Decrease storefront header height.
2. Decrease storefront main navigation height.
3. Add a widget area beside the site logo. I want this widget area to contain a search bar “preferably woocommerce predictive search plugin widget” where the search bar starts from the right end of the logo until the end of the header.
4. Add widget area to storefront main navigation beside other menu items.