The answer to both questions is no, though technically you could do the second.
As Jim said, you can check your server logs (or use usage tracking software/scripts) to determine what referring url a vistor used to get to your site. If the referer is empty, they either typed in your URL or they accessed your from their bookmarks. There is no way of telling what kind of link (ie. image or text) was used to nav. to you. More to the point, who really cares whether someone is using an image or text to link to you? They're linking to you..... that's all that's really important in the situation.
As for loading your page inside an <img> tag, you could possibly do it by either keeping screenshots of your site or using GD to dynamically generate a graphical representation of the site and modifying your output headers to convince the browser to load the image. However, in practise this would be basically useless. None of your links would work (it's only a screenshot of your site, not the real site remember) and the image would need to be at least 640 x 480 to resolve decently. Just about the only way to do what I'm assuming you're think of it to use frames which pull content from your site to the linking site.
geoff