I would like to redesign a random website and use it in my portfolio, either as an image or sample live project. Am I allowed to do that, use their logo and link to the old website, that's still active (just so one can compare two designs)? I'm not going to earn any money (unless the website owner somehow contacts me and is willing to purchase the design).
Duplicate web sites can actually lower SEO. You'd be causing harm to the existing site if you were to duplicate the content and create a separate live web site. I'd recommend you not do this.
If you wish to create an image of a possible redesign, then that would be acceptable because images can't be spidered for content the way actual web sites are. I would recommend that any image you post indicates it's speculative work and not actual live construction. An image can greatly show off your sense of design, but without a live, working site, it's all kind of conjecture since live sites have technical issues which have to be addressed.