"My own take" of an already made site?

From my research in the past, the common consensus is that it is a form of trademark infringement to use their logos and so on. If they become aware of it, they'll send a cease and desist letter which requires you take it down or face legal action, which they are required to do by law. Also, you risk offending them, others they associate with, and even just viewers of your work as potential clients, simply by doing so without their permission to do so. While you might be able to get around certain things with loopholes and technicalities, and even though it might fall into the realm of fair use, or "fan art" in some ways, it's just a bad practice to get into. You can easily "mimic" a website you like and find public domain artwork, logos, and so on to take the place of any copyright or trademark bearing items that would otherwise be present. You can even mention "this is my take on _____" if you feel it necessary but, ultimately, if you are designing something unique anyway, it's better to just let it carry its own merit. People will establish meaningful connections between your work and work they have seen which is similar, all on their own, anyway. In practice, the only reason you should ever directly use their own logos, graphics, and so on would be after you've made contact with them and approached them to see if they'd be willing to entertain a potential redesign of their website in an effort to pick them up as a client.