Apart from the legal issues..
I'd like to create a website that is the exact copy of another website (I do not own but I know will let me do this) where I change a few things, mainly swap images and hide some divs..
So my "website".. how should I build it? I thought something like an Iframe and javascript but I have no idea if this makes sense.. any idea?
In your browser you can choose "Save as HTML" from the File-menu (or how it is called in your browser). You can try different browsers to see which one works best.
Save the web page to your hard disk. There you can make modifications with a HTML-editor. Then you can upload it to your server.
Make sure you have explicit permission.
An iframe and javascript (to do DOM manipulation of the content of the frame) would only work if your site is on the same domain as the site you want to copy, due to the "same origin" policy that browsers implement for security purposes.
As @nob suggests, the workaround is to use some kind of proxy approach, where code running on your own server goes out to the site you want to copy and downloads it, and then passes the content on to your page. You can then either alter the content on your server (via whatever server-side language you are using such as Java or PHP) or, as you suggest, could put the content into a page served from your own server and then manipulate it using Javascript.