If I comment on a Facebook website, can non-friends who 'like' that site see my whole profile?
Only if your Facebook privacy settings are set to allow non-friends to see your whole profile. It is not affected by whether the other person Likes (is a fan of) the page. You can find out what your profile looks like to non-friends by going to Account » Privacy Settings » Basic Directory Information - View settings » Preview My Profile. (If you have permitted some information to be shown to friends-of-friends or networks then people in those categories will see the additional information.)
People will be able to click on your name or picture in your comment (if you haven't explicitly blocked that person), but they will see the same partial profile for you regardless of whether they Like the page. Of course, people who Like the page are more likely to see your comment but anyone could potentially visit the page. If you have turned off your visibility in search so that people cannot find you, someone that may not have been able to find you on Facebook previously may discover you when they see your comment.