Hello,
1) Switcher on the demo is a simple drop-down menu just to show that you can add such content to top bar area. If your plugin has additional options to customize it use options if not then you can do this only by additional customization.
2) We checked that even with the default theme the cart link for example does not redirect to correct EN version https://prnt.sc/100w14k Could you contact Polylang support and ask on how to set up plugin to make it work with the WooCommerce elements in header.
I suppose you need to use Polylang for WooCommerce https://polylang.pro/downloads/polylang-for-woocommerce/
Regards