Should I be citing wrong papers and preprints whose correctness I am unable to judge? I am currently writing a paper with a rather extensive literature trail. There is a question (see bold) that I came across in multiple instances and I can't seem to find an answer to it on this site. Here are two examples.

Firstly, I am unsure whether I should cite papers that are wrong. For example there is a paper which discusses an issue relevant to a part of my paper. However, important parts of the paper have been shown to be wrong, and there is a published comment addressing these issues. A retraction was not found to be necessary.

Secondly, I am citing a couple of preprints, which I studied carefully and found to be relevant for completeness of the discussion. However, while I am able to judge the scope and relevance of the claims, I am unable to judge their correctness, since the methods lie outside my field of expertise.

In both cases above, my main question is whether by citing an article, do I obtain a responsibility for its correctness? I understand that this probably depends on how exactly the work is cited. For definiteness, let us take


  A related model was discussed in [preprint citation].


as an example.

In such a context, the paper is tangentially relevant and should in my opinion be cited, even though it is only for completeness reasons. It is however not peer-reviewed and I am unable to judge its correctness. At the same time I am not stating this anywhere, which would sound weird and obscure the point. So the question here is if I can cite it without being partially responsible if the cited paper turns out to be wrong in the end.

I am aware of this post: Is it right to cite a retracted research article?, however I believe that it discusses a different issue. In this question I am specifically asking about the responsibility of assuring the correctness of cited work, also having in mind that it probably is practically impossible to check all the references in a paper for full correctness.
To answer your general question, I would argue that it is only your responsibility to:


ensure the correctness of your own work and claims;
honestly report your methodology, experiments, and findings;
do a thorough literature review and acknowledge all relevant prior work; and
where appropriate, provide your critical assessment of those works.


Thus, my answer is: no, you are not responsible for whether the cited work is correct.
That being said, you are responsible for making sure that everything that you claim to be true is true.

Regarding your first example

If there is work that has been shown to be inaccurate, as in your first example, you should state as much, and ideally give a citation to the correction.
Depending on how relevant the inaccuracy is to your work, you might explicitly address it in the text, or just acknowledge it in a footnote.
For example:


  Previous work has shown that this issue is important [paper].\footnote{Note that this paper was found to contain errors [correction], but those errors are not relevant to the present discussion.}


Regarding your second example

In the scenario you presented, you are actually stating that the preprint has not been peer-reviewed simply by citing it as such.
In your list of references, it will become clear to the reader that the work you are citing is just a preprint, and therefore has not been peer-reviewed.

However, as mentioned above, it is your responsibility to make sure that anything that you say is accurate.
For example, rather than saying


  Pi is exactly equal to 22/7, as was found previously [preprint citation].


you should say


  A previous study found that pi is exactly equal to 22/7 [preprint citation].


Similarly, in the example you gave, your claim is actually true:


  A related model was discussed in [preprint citation].


(Assuming the preprint does in fact discuss a related model.)
If you instead want to make a claim regarding whether that model is accurate, then you should state what you know to be true, and acknowledge any genuine concerns/skepticism you might have.
For example:


  A previous study showed a similar model to achieve 99.93% accuracy [preprint citation], although that model was only tested against simulated data, so we expect the performance of that model to suffer in the real world.


However, since you note that you are "unable to judge their correctness", I assume you do not have any specific concerns about the cited preprint.
In that case, my advice is to just ensure that your claims are accurate.