" is it possible to have the cart icon visible only in pages that are directly related to the shop "
This is possible.
In the following Link on the Documentation page, Armin has created a tutorial on how to apply CSS (stylings) on specific pages. In your case you wish for only shop related pages.
https://laytheme.com/documentation.html#custom-css-styling
The Icon showing on Mobile is likely a small bug that needs to be remedied.
As Laytheme only just released Woocommerce support, over the next months hopefully all common Bugs will be identified & fixed.
Hope this helps I & thank you for supporting Laytheme - have a wonderful day 🌝
Sincerely Richard