You can make a shortcut to a browser bookmark, just go to your bookmarks, tap & hold, and select 'add to home screen' (or similar, depending on which browser you use). If you just want a link to a browser, just tap and hold the space on the home screen where you want the icon, and select 'add app' (or similar, depending on android version) and pick your installed browser.
If you want to change the icon for a given app, you'll need to install one of the variety of third-party tools for this from Play.
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