Should you mention previous rejections when resubmitting a paper? What is common practice?

Regarding your arguments for: 1. and 3. seem to be more like "axioms" than arguments. 1. in particular seems almost to be begging the question. As for 3., journals have (increasingly!) long lists of requirements for authors, some of them serious but also others that those "in the know" know that they may safely ignore: e.g. many journals ask that you submit papers in their own template, but there's no good reason for that, most papers that I've written or refereed do not do that, and no one cares. So the negation of 3. seems more reasonable to me: if journals wanted you to provide this information, they would ask for it. (Maybe you are trying to argue that they should ask for it. Interesting, but a different question, I think.)

Argument 2. is much more pertinent: if the referee's careful opinion is that your paper is valuable and deserves to be published in a research journal but just not in the journal that you have submitted to, one can indeed save a lot of people's time by carrying over the same referee to a different journal. If you want to do this, it seems best to ask the editor of journal A to get in touch with journal B. I've done this both as an author and as the carry-over referee.

But this is an option that the author gets to choose to exercise or not. In terms of the current practices, this is certainly true and I don't think anyone expects otherwise. Again, it is interesting to argue about whether it should be true....and yes, I think it should. The refereeing system is not strong on incorporating "responses to referees": when authors write back to the editors pointing out factual mistakes in a referee report, they are often told something like "I am inclined to believe you and that is most regrettable. You should definitely resubmit your paper to another journal of similar stature. Best of luck." Very often the referee acts not only as the jury but also the judge and the executioner in the current system. But for a paper whose work is agreed to be correct and novel -- or for which this evaluation was not even made -- then fundamentally, "not good enough for a journal like X" is nothing but an informed opinion. If that opinion is a sound one, a different referee will probably come to it independently. If it isn't, then I think the author has every right to start out fresh.

Let me also say that as a frequent referee, I do not feel the desire to know the provenance of the paper, and I don't think that would help me with my decision. I already have what I think is probably too much information at my disposal: namely the author's identity and thus their professional reputation, my personal view of them, and so forth. Being a good referee doesn't mean taking all possible information into account; it means taking exactly the right information into account to evaluate the paper on its own merits.

In the course of arguing against the "arguments for", it seems that I have espoused some of the arguments against: 2. and 3. I hope these arguments have been shown to be more convincing. (Arguments against 1. and 4. do not seem very compelling to me.)

You should not mention previous rejections. You should definitely revise and take into consideration the comments from the reviewers in Journal A.

I think it would be silly to simply resubmit the identical article at a different journal (unless it is the case mentioned below by @user11192 that you received a rejection with NO feedback), and if you are submitting a revised article then the critiques of the previous reviewer are no longer relevant!

Of course, if you are resubmitting, then you believe that your work is an original contribution. If the journal A reviewer simply didn't like your writing style or you didn't effectively communicate your work, then resubmitting is quite common. If the journal A reviewer pointed out that your work was unoriginal and had been published previously by someone else, then it is simply unethical to attempt to re-submit. In the more subtle case that the first reviewer felt that your contribution was not "substantial" enough, then you still want to revise, and may just end up changing your introduction/motivation more than your core work.

Many journals ask for information on previous submissions their electronic submission systems along with asking if the manuscript is under consideration elsewhere. The fact that some journals have rejection rates way in excess of 50% means that rejections are not necessarily terrible and down-grading your work. Part of rejections are also because the material may not be suitable for the journal in question, which involves no judgement of the quality of your science.

So, if your paper has been rejected as not suitable for the intended journal or some other technical reason, I see no reason to mention this at a resubmission. You made a mistake, full stop.

If the paper has been rejected for some scientific reason, particularly if it has been through review I think mentioning the history is worthwhile. There is of course no law that states you must.

The importance of mentioning the history of a paper is not so much to convey its rejection but to provide the editor with information on why it was rejected and what you took away from this to improve your manuscript. I very much doubt editors will contact other editors to check on previous submissions, none of the parties typically have the time to spare. Instead it is your description of why you think the manuscript should be published, the value of your science, and how you have improved it which is of use to an editor.

I have found out by accidentally assigning the same reviewers to manuscripts, submitted as new, that they have been considered elsewhere. As an editor, this raises a warning flag for me: "is the manuscript so poor/controversial/other that the author wants to hide its past"? The fact that people prefer to send material to certain journals before others is known to most of us. Editors are usually scientists that make the same choices, except perhaps to send it to their own journal.

I therefore think that being open about the problems can help. Of course,if your manuscript is poor, it just is; but if it can and has been improved, rejection by another journal should not stand in your way to get it published elsewhere.

Regarding reviewers, there are two form of replies I usually get when accidentally assigning the same reviewer as for a previous submission: (1) don't want to touch it again, or (2) want to see how it has been improved. The fact that a manuscript was rejected does not mean reviewers were dead against its content. Or, perhaps, some where but for reasons disguised as scientific. I therefore think it is also useful to know something about the previous process when a paper is resubmitted. the place for this is in the accompanying submission letter. Just make it brief but clear.