Kiwi,
No, you can't replace the Like button. You can only use the styles created by Facebook: standard, box-count, and button-count. Facebook doesn't allow custom images because admins would undoubtedly use them to 'trick' users to click them (Click here and get a coupon!) or something. Facebook wants all Likes to be real, organic Likes, and they must use Facebook's recognized branding of the Like button to ensure consistency.
The Fanbox like count is for your Facebook page:
facebook.com/YourPage. The Like module can be for your Fanpage if you set the URL to your Page, or it can be for URLs on your site. Each URL on your site can have a completely different # of Likes, and that's perfectly fine, or you could set the URL to always be your home page (to consolidate Likes) or your Fan Page or all 3. It's your choice.
Alex