In practice, people often just put a "draft" of the paper on their personal website without telling their publisher.

Papers in fancy journals are more prestigious, but the papers that get cited the most are the ones with full text freely available on the Web. So, when they can, scientists choose both.

(The better way to do this is to support a publisher that's both high-quality and open-access and submit to their journal.)

It depends on the venue. In computer science, it is often possible, but there are often weird restrictions: embargo delay, only the preprint, must include a link to the publisher version, only on a personal webpage or on certain kinds of repositories, copyright assignment has been signed as well, etc. So you don't always feel comfortable doing it.

Further, we academics are incentivized, in terms of career, to publish in prestigious venues managed by official publishers. We aren't incentivized to publish in open access repositories as well. So, even though most of us want our articles to be disseminated widely, not all of us can find the time to make sure that the paper is also available on an open repository. This is a factor that explains why many articles are not available for free online even in cases where it would be legal for researchers to upload them.

It depends on the venue.

In my field of computer science (visual analytics / visualizations) at least some of the leading journals/conferences officially allow you to put a version of the paper on your website.

Just today we submitted a paper where the copyright form explicitly mentions that.

It depends upon your field and your country but many journals require a copyright assignment. They also often require that the work has not been published elsewhere previously.