U.S. copyright law is a complicated mix of grey areas and isn't always consistant or logical. The answers seem to vary based on content (music vs film vs tv vs books vs software) But most of the questions you asked have clear answers.
Let me answer your questions in three parts. The Betamax Case (Sony Corp. of America v. Universal City Studios, Inc.) was a US Supreme Court decision that allows people to "time-shift" their television programming. This is why it is legal to use DVRs, VCRs, and video capture cards on your PC for time-shifting purposes. This case means that TiVo does not have to pay royalties to the networks. However, TiVo is starting to do streaming / video-on-demand which likely has a very different arrangement with the networks.
If you "own a season of something" (like on DVD) you are not allowed to make a digital copy of it on your computer. Although this may seem like Fair Use, it is trumped by the Digital Millennium Copyright Act (DCMA). To do so would require you to circumvent the copyright protection on the DVD thus breaking the DCMA law. If by "own a season" you mean have it stored on your TiVo, you are allowed to keep a copy on your computer because you're not circumventing copy protection and you are time-shifting for private use. If you are talking about other content (music and software) the laws are different and allow for a "fair use" backup copy.
Posting a show on the internet has not been covered by the Betamax case. It isn't really time-shifting for private use. The person who posted is clearly not acting in accordance with the laws. The website hosting the video is in a legal grey-area (still wrapped up in the courts). The person who downloads the TV show is also in a legal grey-area (law may be in place but not enforced). Unlike music or films, I haven't heard of anyone prosecuted for downloading TV shows. Now days, this is becoming less of an issue because the networks are putting most of their shows online anyways.