You can edit this in your theme editor.
I'd probably start by reading theme.liquid Layout to see where things are inserted, I'm guessing you'll end up in a Section called header.liquid, or maybe a Snippet with a similar name.
Yes you can change the look of the cart page. Start by looking in the Template called cart.liquid, in your theme editor.
If you're not familiar with editing themes, here's a starter guide I wrote up: https://speedboostr.com/how-to-safely-edit-your-shopify-theme.
If you prefer to hire for development, give a shout to my team at speedboostr.com/contact. We do stuff like this every day.
To change the look of the /cart you would customized the cart.liquid template or similar.
As for changing an icons url , that depends on where the icon is be it in a sidebar, a flyout menu or a static header on everypage.
If it's in the header often this is in file name header.liquid, or it's part of theme.liquid
In more modern themes you'll want to look for routes.cart_url
Otherwise it should simply be an anchor tag that uses /cart in it's href attribute.
If you need this fixed contact me at firstname.lastname@example.org with store url,theme name and details.