Is strict adherence to a citation format really necessary in actual research?

First of all, almost nobody formats their references by hand. Typesetting systems such as Latex have a tool for auto-generating references, for instance from a bibtex file (which is essentially a local database of papers that the researcher often cites). Entries into this file are also not really manual in most cases, as bibtex entries for papers can usually be obtained via a quick web search from the publisher. For instance, in the ACM Digital Library (see example entry here), you can click on the right-hand side on Export -> Bibtex and obtain a bibtex entry. Other typesetting systems, such as Microsoft Word, usually have similar plugins or features.

Now, why would your English teacher (or publishers) even care about whether you write Thomasz Antonie or T. Antonie (or Antonie, T., ...)? At the end of the day, this is not so much about understandability of the reference, but more about giving the paper a more professional look-and-feel. For journals it is important to retain their "trademark" optics. This includes obvious aspects, such as font size, column layout, or margins, but also extends to using a consistent reference style within each paper, and also across papers published in the same journal.

No. Most people use automatically generated citations and references. Usually I find some errors (often case errors) and fix them. But strict adherence to a citation style is typically obtained by the copy editor of the journal, or nor at all.

In my field (mathematics), citation style is almost irrelevant. Journals have house styles and most people more or less try to follow them. But I just can't imagine a paper being rejected for citation style issues. Sometimes a copy editor will correct the citations; sometimes not. As a referee I have occasionally complained about lack of citations for material that the authors take for granted as background knowledge, and I have occasionally suggested corrections to the spelling (especially capitalisation) of some cited titles. But given the often illogical and internally inconsistent nature of citation styles I would likely just ignore even relatively blatant inconsistencies.

People tend to be a bit more careful when writing a thesis than for research articles, and of course professionally published books normally receive a thorough copy editing anyway.

I am pretty sure things are quite similar in computer science and physics. At the opposite end, a lot of people in the humanities seem to be obsessed about this kind of thing. I suspect this is related to the fact that they typically have no way of checking whether a claim is objectively true or objectively false. A lot is essentially a matter of opinion. As a result, typical authorities in those fields are never wrong, or rather, would never admit to being wrong. Since walking off in the wrong direction and changing course when you realise it is a key part of progress, this is a very unhealthy atmosphere that tends to favour mediocre, anti-innovative people who obsess over trivial details that are objectively decidable, so that they find (trivial but verifiable) errors in other people's work while making sure they can't be attacked in this way by their kind.

All that said, there are good reasons for citation styles and one shouldn't just ignore them. In particular, since URLs have a habit of becoming invalid after a few years, just naming author and URL (as in the fourth example) is not a good idea. It's bad enough that cited websites can change their content, but with so little information it can even be hard to track down the contents of one that has disappeared.