My question is whether preprints continue to exist after the information contained in them gets published in a journal. It would be confusing to have two sources for the same information (the preprint and the official publication), so I am assuming that preprints are taken down by the authors after their work gets published in a Journal. My question also is concerned with their citations. If preprints and peer-reviewed articles exist, I don't know if tools such as Google Scholar are adding the total number of the citations received by both papers (I believe they should, as they are practically the same publication, only with different levels of revisions).
In general, preprints continue to remain available even after publication. In fact, on most preprint servers, such as arXiv, it is impossible to completely remove a preprint once submitted, and any updates are stored as new versions. The preprint itself is a form of publication, where you can disseminate results, get feedback, and get a time stamp for your work without waiting for a lengthy peer review process.
Preprints may be cited and used to derive other works before they are published, so it is important to preserve past versions for the references to be coherent, particularly since a paper may be updated several times before publication in a peer-reviewed venue. The preprint version may be more likely to receive corrections and updates after publication, since that is easier than updating the journal version. It may also contain extra supplementary material missing from the published version due to length constraints, and so it is not necessary that the two versions are identical.
Regarding citations, services such as Google Scholar are usually able to interpret several versions of the same paper as a single entry. However, issues with this do exist, and have so far not been satisfactorily resolved.
This is a genuine problem with the old-times notion of "publication", that is, in static (peer-reviewed or not) forms.
The so-called "preprint" forms on peoples' home pages or arXiv and such can be updated, corrected, and so on.
The traditional form of citations has the analogous problem, of having been developed gradually in a time when there was no internet, so people couldn't communicate easily themselves, ... couldn't even typeset things themselves. In those times, a "reference" was an utterly fixed, static thing, which would never change... and its deficits were fixed in stone forever, etc.
So it's not a surprise that the old citation concept cannot cope with more dynamic references, locations, and so on. Don't over-think how to cope with such a failure... nor believe that there should be a real "solution" in the old context.
Preprints, in the form of pre-drafts ("preliminary and incomplete"), working papers, under submission versions etc, remain in most cases available online. This is, in my opinion, a good thing, for many reasons.
Influential working papers may remain as such for many years, while gathering hundreds, if not thousands, of citations. My favourite example is a very extensive, heavily methodological piece that was put online in 2002 and got an official reference in 2009, as a book chapter. If the old version was removed, the old citations might not be tractable.
It can be both useful and interesting to track the evolution of a draft. Sometimes, interesting footnotes, quotes or even sub-sections are not in the published paper (e.g. after following reviewer recommendations) but can still be useful to someone else. The same holds for changes, mistakes etc.
EDIT - I forgot the most important point, mentioned by @BruceET : a publicly available copy, almost identical to the published work, that is not behind a paywall. This is by far the best reason.
I am not aware of a working paper being updated after the work is published, but it might contain more detailed tables, appendices etc.